Theme Song
# install.packages("remotes")
library('BBmisc', 'rmsfuns')
#remotes::install_github("rstudio/sass")
lib('sass')
## sass
## TRUE
/* https://stackoverflow.com/a/66029010/3806250 */
h1 { color: #002C54; }
h2 { color: #2F496E; }
h3 { color: #375E97; }
h4 { color: #556DAC; }
h5 { color: #92AAC7; }
/* ----------------------------------------------------------------- */
/* https://gist.github.com/himynameisdave/c7a7ed14500d29e58149#file-broken-gradient-animation-less */
.hover01 {
/* color: #FFD64D; */
background: linear-gradient(155deg, #EDAE01 0%, #FFEB94 100%);
transition: all 0.45s;
&:hover{
background: linear-gradient(155deg, #EDAE01 20%, #FFEB94 80%);
}
}
.hover02 {
color: #FFD64D;
background: linear-gradient(155deg, #002C54 0%, #4CB5F5 100%);
transition: all 0.45s;
&:hover{
background: linear-gradient(155deg, #002C54 20%, #4CB5F5 80%);
}
}
.hover03 {
color: #FFD64D;
background: linear-gradient(155deg, #A10115 0%, #FF3C5C 100%);
transition: all 0.45s;
&:hover{
background: linear-gradient(155deg, #A10115 20%, #FF3C5C 80%);
}
}
## https://stackoverflow.com/a/36846793/3806250
options(width = 999)
knitr::opts_chunk$set(class.source = 'hover01', class.output = 'hover02', class.error = 'hover03')
if(!suppressPackageStartupMessages(require('BBmisc'))) {
install.packages('BBmisc', dependencies = TRUE, INSTALL_opts = '--no-lock')
}
suppressPackageStartupMessages(require('BBmisc'))
# suppressPackageStartupMessages(require('rmsfuns'))
pkgs <- c('devtools', 'knitr', 'kableExtra', 'tidyr',
'readr', 'lubridate', 'data.table', 'reprex',
'timetk', 'plyr', 'dplyr', 'stringr', 'magrittr',
'tdplyr', 'tidyverse', 'formattable',
'echarts4r', 'paletteer')
suppressAll(lib(pkgs))
# load_pkg(pkgs)
## Set the timezone but not change the datetime
Sys.setenv(TZ = 'Asia/Tokyo')
## options(knitr.table.format = 'html') will set all kableExtra tables to be 'html', otherwise need to set the parameter on every single table.
options(warn = -1, knitr.table.format = 'html')#, digits.secs = 6)
## https://stackoverflow.com/questions/39417003/long-vectors-not-supported-yet-abnor-in-rmd-but-not-in-r-script
knitr::opts_chunk$set(message = FALSE, warning = FALSE)#,
#cache = TRUE, cache.lazy = FALSE)
rm(pkgs)
課題をすぐに提出してください
課題の提出期限は、5月31日 15:59 JSTですが、可能であれば1日か2日早く提出してください。 早い段階で提出すると、他の受講生のレビューを時間内に得る可能性が高くなります。
The R dataset faithful contains data on waiting time between eruptions (the column named waiting) and the duration of the eruption (the column named eruptions) for the famous Old Faithful geyser in Yellowstone National Park, Wyoming, USA.
In this case, you are asked to modify the MCMC algorithm provided in “Sample code for density estimation problems” (as opposed to the EM algorithm you used in the previous peer assignment) to provide a (Bayesian) density estimate the marginal distribution of the duration of the eruptions using a location-and-scale mixture of 2 univariate Gaussian distributions (as opposed to the location mixture of 6 univariate Gaussian distributions that we used for the galaxies dataset). Assume that the priors are \(\omega∼Beta(1,1)\), \(\mu_k∼Normal(η,\tau^2)\) and \(1/σ^2_k∼Gamma(d,q)\), where \(\eta\), \(\tau^2\), \(d\) and \(q\) are selected using an empirical Bayes approach similar to the one we used in “Sample code for density estimation problems”.
Reviewers will check whether the code has been modified correctly, and whether the density estimate you generate appears correct. Please remember that you are being asked to use a location-and-scale mixture to generate the density estimate, so the “Sample code for density estimation problems” cannot be used directly and requires some modification. Before submitting your answer, it might be useful to compare the density estimate generated by your algorithm against a kernel density estimate generated by the R function density(), and agains the answer to the previous peer assignment.
Provide an MCMC algorithm to fit the two-component, location-and-scale mixture of Gaussians.
## Load package
if(!suppressPackageStartupMessages(require('invgamma'))) {
##install.packages('invgamma', dependencies = TRUE, INSTALL_opts = '--no-lock')
devtools::install_github('dkahle/invgamma', force=TRUE)
}
## these few packages have `rinvgamma` function.
#library('EDISON', 'extraDistr', 'MCMCpack')
## function in 'invgamma' will be more accurate than 'MCMCpack'
suppressPackageStartupMessages(require('invgamma'))
suppressAll(lib('EDISON', 'extraDistr', 'MCMCpack'))
### Get a "Bayesian" kernel density estimator based on the same location mixture of 6 normals
## Priors set up using an "empirical Bayes" approach
KK = 2
aa = rep(1,KK)
eta = mean(x)
## Error in mean(x): object 'x' not found
tau = sqrt(var(x))
## Error in is.data.frame(x): object 'x' not found
dd = 2
qq = var(x)/KK
## Error in is.data.frame(x): object 'x' not found
mu_0 = rnorm(KK, eta, tau)
## Error in rnorm(KK, eta, tau): object 'eta' not found
sigma_0 = invgamma::rinvgamma(KK, dd, qq)
## Error in rgamma(n, shape, rate): object 'qq' not found
## Initialize the parameters
w = rep(1,KK)/KK
mu = rnorm(KK, mean(x), sd(x))
## Error in mean(x): object 'x' not found
sigma = sd(x)/KK
## Error in is.data.frame(x): object 'x' not found
cc = sample(1:KK, n, replace=T, prob=w)
## Error in sample.int(length(x), size, replace, prob): invalid 'size' argument
## Number of iterations of the sampler
rrr = 12000
burn = 2000
## Storing the samples
cc.out = array(0, dim=c(rrr, n))
## Error in array(0, dim = c(rrr, n)): 'list' object cannot be coerced to type 'integer'
w.out = array(0, dim=c(rrr, KK))
mu.out = array(0, dim=c(rrr, KK))
sigma.out = rep(0, rrr)
logpost = rep(0, rrr)
for(s in 1:rrr){
# Sample the indicators
for(i in 1:n){
v = rep(0,KK)
for(k in 1:KK){
v[k] = log(w[k]) + dnorm(x[i], mu[k], sigma, log=TRUE) #Compute the log of the weights
}
v = exp(v - max(v))/sum(exp(v - max(v)))
cc[i] = sample(1:KK, 1, replace=TRUE, prob=v)
}
# Sample the weights
## https://cran.r-project.org/web/packages/rBeta2009/rBeta2009.pdf
## gtools, extraDistr, MCMCpack also have rbeta and rdirichlet functions.
w = as.vector(rdirichlet(1, aa + tabulate(cc, nbins=KK)))
# Sample the means
for(k in 1:KK){
nk = sum(cc==k)
xsumk = sum(x[cc==k])
tau2.hat = 1/(nk/sigma^2 + 1/sigma_0[k]^2)
mu.hat = tau2.hat*(xsumk/sigma^2 + mu_0[k]/sigma_0[k]^2)
mu[k] = rnorm(1, mu.hat, sqrt(tau2.hat))
}
# Sample the variances
dd.star = dd + n/2
qq.star = qq + sum((x - mu[cc])^2)/2
sigma = sqrt(1/rgamma(1, dd.star, qq.star))
# Store samples
cc.out[s,] = cc
w.out[s,] = w
mu.out[s,] = mu
sigma.out[s] = sigma
for(i in 1:n){
logpost[s] = logpost[s] + log(w[cc[i]]) + dnorm(x[i], mu[cc[i]], sigma, log=TRUE)
}
logpost[s] = logpost[s] + log(ddirichlet(w, aa))
for(k in 1:KK){
logpost[s] = logpost[s] + dnorm(mu[k], eta, tau, log=TRUE)
}
logpost[s] = logpost[s] + dgamma(1/sigma^2, dd, qq, log=TRUE) - 4*log(sigma)
if(s/500==floor(s/500)){
print(paste("s =",s))
}
}
## Error in 1:n: NA/NaN argument
Provide code to generate the density estimate on a grid consisting of 150 points in the interval [0,7].
xx = seq(0,7,length.out=150)
nxx = length(xx)
## Compute the samples of the density over a dense grid
density.mcmc = array(0, dim=c(rrr-burn,length(xx)))
for(s in 1:(rrr-burn)){
for(k in 1:KK){
density.mcmc[s,] = density.mcmc[s,] + w.out[s+burn,k]*dnorm(xx,mu.out[s+burn,k],sigma.out[s+burn])
}
}
density.mcmc.m = apply(density.mcmc , 2, mean)
Provide the a graph of the density estimate on the interval [0,7].
## Plot Bayesian estimate with pointwise credible bands along with kernel density estimate and frequentist point estimate
colscale = c("black", "blue", "red")
yy = density(x)
## Error in density(x): object 'x' not found
density.mcmc.lq = apply(density.mcmc, 2, quantile, 0.025)
## Error in quantile.default(newX[, i], ...): missing values and NaN's not allowed if 'na.rm' is FALSE
density.mcmc.uq = apply(density.mcmc, 2, quantile, 0.975)
## Error in quantile.default(newX[, i], ...): missing values and NaN's not allowed if 'na.rm' is FALSE
plot(xx, density.mcmc.m, type="n",ylim=c(0,max(density.mcmc.uq)),xlab="Eruption Duration", ylab="Density")
## Error in plot.default(xx, density.mcmc.m, type = "n", ylim = c(0, max(density.mcmc.uq)), : object 'density.mcmc.uq' not found
polygon(c(xx,rev(xx)), c(density.mcmc.lq, rev(density.mcmc.uq)), col="grey", border="grey")
## Error in xy.coords(x, y, setLab = FALSE): object 'density.mcmc.lq' not found
lines(xx, density.mcmc.m, col=colscale[1], lwd=2)
## Error in plot.xy(xy.coords(x, y), type = type, ...): plot.new has not been called yet
lines(xx, density.EM, col=colscale[2], lty=2, lwd=2)
## Error in xy.coords(x, y): object 'density.EM' not found
lines(yy, col=colscale[3], lty=3, lwd=2)
## Error in lines(yy, col = colscale[3], lty = 3, lwd = 2): object 'yy' not found
points(x, rep(0,n))
## Error in points(x, rep(0, n)): object 'x' not found
legend(5, 0.45, c("KDE","EM","MCMC"), col=colscale[c(3,2,1)], lty=c(3,2,1), lwd=2, bty="n")
## Error in strwidth(legend, units = "user", cex = cex, font = text.font): plot.new has not been called yet
Is the sampler for \(c\) correct?
The full conditional for \(c_i\) is given by,
\[\begin{align} \Pr(c_i = 1|...) = \frac{\omega\frac{1}{\sqrt[]{2\pi\sigma_1}}exp\left\{-\frac{1}{2\sigma^2_1}(x_i-\mu_1)^2\right\}}{\omega\frac{1}{\sqrt[]{2\pi\sigma_1}}exp\left\{-\frac{1}{2\sigma^2_1}(x_i-\mu_1)^2\right\}+(1-\omega)\frac{1}{\sqrt[]{2\pi\sigma_2}}exp\left\{-\frac{1}{2\sigma^2_2}(x_i-\mu_2)^2\right\}} \end{align}\]
(This assumes that the mixture weights are parameterized as
for(i in 1:n){
v = rep(0,2)
v[1] = log(w) + dnorm(x[i], mu[1], sigma[1], log=TRUE) #Compute the log of the weights
v[2] = log(1-w) + dnorm(x[i], mu[2], sigma[2], log=TRUE) #Compute the log of the weights
v = exp(v - max(v))/sum(exp(v - max(v)))
cc[i] = sample(1:2, 1, replace=TRUE, prob=v)
}
## Error in 1:n: NA/NaN argument
Is the sampler for \(\omega\) correct?
The full conditional is simply
\[ \omega \mid \cdots \mbox{Beta} \left(\alpha_1 + \sum_{i=1}^{n} \mathbf{1}(c_i=1), \alpha_2 + \sum_{i=1}^{n} \mathbf{1}(c_i=2)\right) \]
The following one line of code implements it:
w = rbeta(1, aa[1] + sum(cc==1), aa[2] + sum(cc==2))
## Error in rbeta(1, aa[1] + sum(cc == 1), aa[2] + sum(cc == 2)): object 'cc' not found
Are the sampler for \(\mu\) correct?
\[ \mu_k \mid \cdots \mbox{Normal}\left( \left[ \frac{n_k}{\sigma_k^2} + \frac{1}{\tau^2} \right]^{-1}\left[ \frac{\sum_{i:c_i=k} x_i}{\sigma_k^2} + \frac{\eta}{\tau^2} \right], \left[ \frac{n_k}{\sigma_k^2} + \frac{1}{\tau^2} \right]^{-1}\right) \]
The following code implements the sampler:
# Sample the means
for(k in 1:2){
nk = sum(cc==k)
xsumk = sum(x[cc==k])
tau2.hat = 1/(nk/sigma[k]^2 + 1/tau^2)
mu.hat = tau2.hat*(xsumk/sigma[k]^2 + eta/tau^2)
mu[k] = rnorm(1, mu.hat, sqrt(tau2.hat))
}
## Error in eval(expr, envir, enclos): object 'cc' not found
Remember, however, that there are various ways in which the implementation could proceed.
Are the samplers for \(\sigma_k\) correct?
The full conditional takes the form:
\[ 1/\sigma^2_k \mid \cdots \mbox{Gamma}\left(d + \frac{n_k}{2} , q + \frac{1}{2} \sum_{i:c_k=k} (x_i - \mu_k)^2 \right) \]
The following code implements the sampler:
# Sample the variances
for(k in 1:2){
nk = sum(cc==k)
dd.star = dd + nk/2
qq.star = qq + sum((x[cc==k] - mu[k])^2)/2
sigma[k] = sqrt(1/rgamma(1, dd.star, qq.star))
}
## Error in eval(expr, envir, enclos): object 'cc' not found
Remember, however, that there are various ways in which the implementation could proceed.
Does the code to generate the density estimate appear correct?
This is an example that, as before, assumes that the weights are parameterized as \(\omega\) and \(1- \omega\).
## Plot Bayesian estimate with pointwise credible bands
density.mcmc.lq = apply(density.mcmc, 2, quantile, 0.025)
## Error in quantile.default(newX[, i], ...): missing values and NaN's not allowed if 'na.rm' is FALSE
density.mcmc.uq = apply(density.mcmc, 2, quantile, 0.975)
## Error in quantile.default(newX[, i], ...): missing values and NaN's not allowed if 'na.rm' is FALSE
plot(xx, density.mcmc.m, type="n",ylim=c(0,max(density.mcmc.uq)),xlab="Eruptions", ylab="Density")
## Error in plot.default(xx, density.mcmc.m, type = "n", ylim = c(0, max(density.mcmc.uq)), : object 'density.mcmc.uq' not found
polygon(c(xx,rev(xx)), c(density.mcmc.lq, rev(density.mcmc.uq)), col="grey", border="grey")
## Error in xy.coords(x, y, setLab = FALSE): object 'density.mcmc.lq' not found
lines(xx, density.mcmc.m, col="black", lwd=2)
## Error in plot.xy(xy.coords(x, y), type = type, ...): plot.new has not been called yet
points(x, rep(0,n))
## Error in points(x, rep(0, n)): object 'x' not found
Note that this will generate not only a point estimate, but also pointwise credible intervals.
Does the graph appear to be correct?
The density estimate should look similar to the Figure below. Note that the variance of both components is clearly different.
The full code for the MCMC algorithm follows:
x = faithful$eruptions
n = length(x)
plot(density(x))
points(x,rep(0,n))
## Priors set up using an "empirical Bayes" approach
aa = rep(1,2)
eta = mean(x)
tau = sqrt(var(x))
dd = 2
qq = var(x)/2
## Initialize the parameters
w = 1/2
mu = rnorm(2, mean(x), sd(x))
sigma = rep(sd(x)/2,2)
cc = sample(1:2, n, replace=T, prob=c(1/2,1/2))
## Number of iterations of the sampler
rrr = 12000
burn = 2000
## Storing the samples
cc.out = array(0, dim=c(rrr, n))
w.out = rep(0, rrr)
mu.out = array(0, dim=c(rrr, 2))
sigma.out = array(0, dim=c(rrr, 2))
logpost = rep(0, rrr)
for(s in 1:rrr){
# Sample the indicators
for(i in 1:n){
v = rep(0,2)
v[1] = log(w) + dnorm(x[i], mu[1], sigma[1], log=TRUE) #Compute the log of the weights
v[2] = log(1-w) + dnorm(x[i], mu[2], sigma[2], log=TRUE) #Compute the log of the weights
v = exp(v - max(v))/sum(exp(v - max(v)))
cc[i] = sample(1:2, 1, replace=TRUE, prob=v)
}
# Sample the weights
w = rbeta(1, aa[1] + sum(cc==1), aa[2] + sum(cc==2))
# Sample the parameters of the components
for(k in 1:2){
# Sample the means
nk = sum(cc==k)
xsumk = sum(x[cc==k])
tau2.hat = 1/(nk/sigma[k]^2 + 1/tau^2)
mu.hat = tau2.hat*(xsumk/sigma[k]^2 + eta/tau^2)
mu[k] = rnorm(1, mu.hat, sqrt(tau2.hat))
# Sample the variances
dd.star = dd + nk/2
qq.star = qq + sum((x[cc==k] - mu[k])^2)/2
sigma[k] = sqrt(1/rgamma(1, dd.star, qq.star))
}
# Store samples
cc.out[s,] = cc
w.out[s] = w
mu.out[s,] = mu
sigma.out[s,] = sigma
logpost[s] = 0
for(i in 1:n){
if(cc[i] == 1){
logpost[s] = logpost[s] + log(w) + dnorm(x[i], mu[1], sigma[1], log=TRUE)
}else{
logpost[s] = logpost[s] + log(1-w) + dnorm(x[i], mu[2], sigma[2], log=TRUE)
}
}
logpost[s] = logpost[s] + dbeta(w, aa[1], aa[2], log=TRUE)
for(k in 1:2){
logpost[s] = logpost[s] + dnorm(mu[k], eta, tau, log = T) + dgamma(1/sigma[k]^2, dd, qq)/sigma[k]^4
}
if(s/500==floor(s/500)){
print(paste("s =",s))
}
}
## [1] "s = 500"
## [1] "s = 1000"
## [1] "s = 1500"
## [1] "s = 2000"
## [1] "s = 2500"
## [1] "s = 3000"
## [1] "s = 3500"
## [1] "s = 4000"
## [1] "s = 4500"
## [1] "s = 5000"
## [1] "s = 5500"
## [1] "s = 6000"
## [1] "s = 6500"
## [1] "s = 7000"
## [1] "s = 7500"
## [1] "s = 8000"
## [1] "s = 8500"
## [1] "s = 9000"
## [1] "s = 9500"
## [1] "s = 10000"
## [1] "s = 10500"
## [1] "s = 11000"
## [1] "s = 11500"
## [1] "s = 12000"
## Compute the samples of the density over a dense grid
xx = seq(0,7,length=150)
density.mcmc = array(0, dim=c(rrr-burn,length(xx)))
for(s in 1:(rrr-burn)){
density.mcmc[s,] = density.mcmc[s,] + w.out[s+burn]*dnorm(xx,mu.out[s+burn,1],sigma.out[s+burn,1]) +
(1-w.out[s+burn])*dnorm(xx,mu.out[s+burn,2],sigma.out[s+burn,2])
}
density.mcmc.m = apply(density.mcmc , 2, mean)
## Plot Bayesian estimate with pointwise credible bands
density.mcmc.lq = apply(density.mcmc, 2, quantile, 0.025)
density.mcmc.uq = apply(density.mcmc, 2, quantile, 0.975)
plot(xx, density.mcmc.m, type="n",ylim=c(0,max(density.mcmc.uq)),xlab="Eruptions", ylab="Density")
polygon(c(xx,rev(xx)), c(density.mcmc.lq, rev(density.mcmc.uq)), col="grey", border="grey")
lines(xx, density.mcmc.m, col="black", lwd=2)
points(x, rep(0,n))
Provide an MCMC algorithm to fit the two-component, location-and-scale mixture of Gaussians.
### Get a "Bayesian" kernel density estimator based on the same location mixture of 6 normals
## Priors set up using an "empirical Bayes" approach
aa = rep(1,KK)
eta = mean(x)
tau = sqrt(var(x))
dd = 2
qq = var(x)/KK
mu_0 = rnorm(KK, eta, tau)
sigma_0 = rinvgamma(KK, dd, qq)
## Initialize the parameters
w = rep(1,KK)/KK
mu = rnorm(KK, mean(x), sd(x))
sigma = sd(x)/KK
cc = sample(1:KK, n, replace=T, prob=w)
## Number of iterations of the sampler
rrr = 12000
burn = 2000
## Storing the samples
cc.out = array(0, dim=c(rrr, n))
w.out = array(0, dim=c(rrr, KK))
mu.out = array(0, dim=c(rrr, KK))
sigma.out = rep(0, rrr)
logpost = rep(0, rrr)
for(s in 1:rrr){
# Sample the indicators
for(i in 1:n){
v = rep(0, KK)
for(k in 1:KK){
v[k] = log(w[k]) + dnorm(x[i], mu[k], sigma, log = TRUE) #Compute the log of the weights
}
v = exp(v - max(v))/sum(exp(v - max(v)))
cc[i] = sample(1:KK, 1, replace = TRUE, prob = v)
}
# Sample the weights
w = as.vector(rdirichlet(1, aa + tabulate(cc, nbins=KK)))
# Sample the means
for(k in 1:KK){
nk = sum(cc==k)
xsumk = sum(x[cc==k])
tau2.hat = 1/(nk/sigma^2 + 1/sigma_0[k]^2)
mu.hat = tau2.hat*(xsumk/sigma^2 + mu_0[k]/sigma_0[k]^2)
mu[k] = rnorm(1, mu.hat, sqrt(tau2.hat))
}
# Sample the variances
dd.star = dd + n/2
qq.star = qq + sum((x - mu[cc])^2)/2
sigma = sqrt(1/rgamma(1, dd.star, qq.star))
# Store samples
cc.out[s,] = cc
w.out[s,] = w
mu.out[s,] = mu
sigma.out[s] = sigma
for(i in 1:n){
logpost[s] = logpost[s] + log(w[cc[i]]) + dnorm(x[i], mu[cc[i]], sigma, log = TRUE)
}
logpost[s] = logpost[s] + log(ddirichlet(w, aa))
for(k in 1:KK){
logpost[s] = logpost[s] + dnorm(mu[k], eta, tau, log = TRUE)
}
logpost[s] = logpost[s] + dgamma(1/sigma^2, dd, qq, log = TRUE) - 4*log(sigma)
if(s/500==floor(s/500)){
print(paste("s =",s))
}
}
## [1] "s = 500"
## [1] "s = 1000"
## [1] "s = 1500"
## [1] "s = 2000"
## [1] "s = 2500"
## [1] "s = 3000"
## [1] "s = 3500"
## [1] "s = 4000"
## [1] "s = 4500"
## [1] "s = 5000"
## [1] "s = 5500"
## [1] "s = 6000"
## [1] "s = 6500"
## [1] "s = 7000"
## [1] "s = 7500"
## [1] "s = 8000"
## [1] "s = 8500"
## [1] "s = 9000"
## [1] "s = 9500"
## [1] "s = 10000"
## [1] "s = 10500"
## [1] "s = 11000"
## [1] "s = 11500"
## [1] "s = 12000"
Provide code to generate the density estimate on a grid consisting of 150 points in the interval [0,7].
xx = seq(0,7,length=150)
nxx = length(xx)
## Compute the samples of the density over a dense grid
density.mcmc = array(0, dim = c(rrr-burn, length(xx)))
for(s in 1:(rrr-burn)){
for(k in 1:KK){
density.mcmc[s,] = density.mcmc[s,] + w.out[s+burn,k]*dnorm(xx,mu.out[s+burn,k],sigma.out[s+burn])
}
}
density.mcmc.m = apply(density.mcmc , 2, mean)
Provide the a graph of the density estimate on the interval [0,7].
## Plot Bayesian estimate with pointwise credible bands along with kernel density estimate and frequentist point estimate
colscale = c("black", "blue", "red")
yy = density(x)
density.mcmc.lq = apply(density.mcmc, 2, quantile, 0.025)
density.mcmc.uq = apply(density.mcmc, 2, quantile, 0.975)
plot(xx, density.mcmc.m, type="n",ylim=c(0,max(density.mcmc.uq)),xlab="Eruption Duration", ylab="Density")
polygon(c(xx,rev(xx)), c(density.mcmc.lq, rev(density.mcmc.uq)), col="grey", border="grey")
lines(xx, density.mcmc.m, col=colscale[1], lwd=2)
#lines(xx, density.EM, col=colscale[2], lty=2, lwd=2)
lines(yy, col=colscale[3], lty=3, lwd=2)
points(x, rep(0,n))
legend(5, 0.45, c("KDE","EM","MCMC"), col=colscale[c(3,2,1)], lty=c(3,2,1), lwd=2, bty="n")
Is the sampler for \(c\) correct?
The full conditional for \(c_i\) is given by,
\[ \Pr(c_i = 1|...) = \frac{\omega\frac{1}{\sqrt[]{2\pi\sigma_1}}exp\left\{-\frac{1}{2\sigma^2_1}(x_i-\mu_1)^2\right\}}{\omega\frac{1}{\sqrt[]{2\pi\sigma_1}}exp\left\{-\frac{1}{2\sigma^2_1}(x_i-\mu_1)^2\right\}+(1-\omega)\frac{1}{\sqrt[]{2\pi\sigma_2}}exp\left\{-\frac{1}{2\sigma^2_2}(x_i-\mu_2)^2\right\}} \]
(This assumes that the mixture weights are parameterized as
for(i in 1:n){
v = rep(0,2)
v[1] = log(w) + dnorm(x[i], mu[1], sigma[1], log=TRUE) #Compute the log of the weights
v[2] = log(1-w) + dnorm(x[i], mu[2], sigma[2], log=TRUE) #Compute the log of the weights
v = exp(v - max(v))/sum(exp(v - max(v)))
cc[i] = sample(1:2, 1, replace=TRUE, prob=v)
}
## Error in sample.int(length(x), size, replace, prob): NA in probability vector
Is the sampler for \(\omega\) correct?
The full conditional is simply
\[ \omega \mid \cdots \mbox{Beta} \left(\alpha_1 + \sum_{i=1}^{n} \mathbf{1}(c_i=1), \alpha_2 + \sum_{i=1}^{n} \mathbf{1}(c_i=2)\right) \]
The following one line of code implements it:
w = rbeta(1, aa[1] + sum(cc==1), aa[2] + sum(cc==2))
Are the sampler for \(\mu\) correct?
\[ \mu_k \mid \cdots \mbox{Normal}\left( \left[ \frac{n_k}{\sigma_k^2} + \frac{1}{\tau^2} \right]^{-1}\left[ \frac{\sum_{i:c_i=k} x_i}{\sigma_k^2} + \frac{\eta}{\tau^2} \right], \left[ \frac{n_k}{\sigma_k^2} + \frac{1}{\tau^2} \right]^{-1}\right) \]
The following code implements the sampler:
# Sample the means
for(k in 1:2){
nk = sum(cc==k)
xsumk = sum(x[cc==k])
tau2.hat = 1/(nk/sigma[k]^2 + 1/tau^2)
mu.hat = tau2.hat*(xsumk/sigma[k]^2 + eta/tau^2)
mu[k] = rnorm(1, mu.hat, sqrt(tau2.hat))
}
Remember, however, that there are various ways in which the implementation could proceed.
Are the samplers for \(\sigma_k\) correct?
The full conditional takes the form:
\[ 1/\sigma^2_k \mid \cdots \mbox{Gamma}\left(d + \frac{n_k}{2} , q + \frac{1}{2} \sum_{i:c_k=k} (x_i - \mu_k)^2 \right) \]
The following code implements the sampler:
# Sample the variances
for(k in 1:2){
nk = sum(cc==k)
dd.star = dd + nk/2
qq.star = qq + sum((x[cc==k] - mu[k])^2)/2
sigma[k] = sqrt(1/rgamma(1, dd.star, qq.star))
}
Remember, however, that there are various ways in which the implementation could proceed.
Does the code to generate the density estimate appear correct?
This is an example that, as before, assumes that the weights are parameterized as \(\omega\) and \(1- \omega\).
## Plot Bayesian estimate with pointwise credible bands
density.mcmc.lq = apply(density.mcmc, 2, quantile, 0.025)
density.mcmc.uq = apply(density.mcmc, 2, quantile, 0.975)
plot(xx, density.mcmc.m, type="n",ylim=c(0,max(density.mcmc.uq)),xlab="Eruptions", ylab="Density")
polygon(c(xx,rev(xx)), c(density.mcmc.lq, rev(density.mcmc.uq)), col="grey", border="grey")
lines(xx, density.mcmc.m, col="black", lwd=2)
points(x, rep(0,n))
Note that this will generate not only a point estimate, but also pointwise credible intervals.
Does the graph appear to be correct?
The density estimate should look similar to the Figure below. Note that the variance of both components is clearly different.
The full code for the MCMC algorithm follows:
x = faithful$eruptions
n = length(x)
plot(density(x))
points(x,rep(0,n))
## Priors set up using an "empirical Bayes" approach
aa = rep(1,2)
eta = mean(x)
tau = sqrt(var(x))
dd = 2
qq = var(x)/2
## Initialize the parameters
w = 1/2
mu = rnorm(2, mean(x), sd(x))
sigma = rep(sd(x)/2,2)
cc = sample(1:2, n, replace=T, prob=c(1/2,1/2))
## Number of iterations of the sampler
rrr = 12000
burn = 2000
## Storing the samples
cc.out = array(0, dim=c(rrr, n))
w.out = rep(0, rrr)
mu.out = array(0, dim=c(rrr, 2))
sigma.out = array(0, dim=c(rrr, 2))
logpost = rep(0, rrr)
for(s in 1:rrr){
# Sample the indicators
for(i in 1:n){
v = rep(0,2)
v[1] = log(w) + dnorm(x[i], mu[1], sigma[1], log=TRUE) #Compute the log of the weights
v[2] = log(1-w) + dnorm(x[i], mu[2], sigma[2], log=TRUE) #Compute the log of the weights
v = exp(v - max(v))/sum(exp(v - max(v)))
cc[i] = sample(1:2, 1, replace=TRUE, prob=v)
}
# Sample the weights
w = rbeta(1, aa[1] + sum(cc==1), aa[2] + sum(cc==2))
# Sample the parameters of the components
for(k in 1:2){
# Sample the means
nk = sum(cc==k)
xsumk = sum(x[cc==k])
tau2.hat = 1/(nk/sigma[k]^2 + 1/tau^2)
mu.hat = tau2.hat*(xsumk/sigma[k]^2 + eta/tau^2)
mu[k] = rnorm(1, mu.hat, sqrt(tau2.hat))
# Sample the variances
dd.star = dd + nk/2
qq.star = qq + sum((x[cc==k] - mu[k])^2)/2
sigma[k] = sqrt(1/rgamma(1, dd.star, qq.star))
}
# Store samples
cc.out[s,] = cc
w.out[s] = w
mu.out[s,] = mu
sigma.out[s,] = sigma
logpost[s] = 0
for(i in 1:n){
if(cc[i] == 1){
logpost[s] = logpost[s] + log(w) + dnorm(x[i], mu[1], sigma[1], log=TRUE)
}else{
logpost[s] = logpost[s] + log(1-w) + dnorm(x[i], mu[2], sigma[2], log=TRUE)
}
}
logpost[s] = logpost[s] + dbeta(w, aa[1], aa[2], log=TRUE)
for(k in 1:2){
logpost[s] = logpost[s] + dnorm(mu[k], eta, tau, log = T) + dgamma(1/sigma[k]^2, dd, qq)/sigma[k]^4
}
if(s/500==floor(s/500)){
print(paste("s =",s))
}
}
## [1] "s = 500"
## [1] "s = 1000"
## [1] "s = 1500"
## [1] "s = 2000"
## [1] "s = 2500"
## [1] "s = 3000"
## [1] "s = 3500"
## [1] "s = 4000"
## [1] "s = 4500"
## [1] "s = 5000"
## [1] "s = 5500"
## [1] "s = 6000"
## [1] "s = 6500"
## [1] "s = 7000"
## [1] "s = 7500"
## [1] "s = 8000"
## [1] "s = 8500"
## [1] "s = 9000"
## [1] "s = 9500"
## [1] "s = 10000"
## [1] "s = 10500"
## [1] "s = 11000"
## [1] "s = 11500"
## [1] "s = 12000"
## Compute the samples of the density over a dense grid
xx = seq(0,7,length=150)
density.mcmc = array(0, dim=c(rrr-burn,length(xx)))
for(s in 1:(rrr-burn)){
density.mcmc[s,] = density.mcmc[s,] + w.out[s+burn]*dnorm(xx,mu.out[s+burn,1],sigma.out[s+burn,1]) +
(1-w.out[s+burn])*dnorm(xx,mu.out[s+burn,2],sigma.out[s+burn,2])
}
density.mcmc.m = apply(density.mcmc , 2, mean)
## Plot Bayesian estimate with pointwise credible bands
density.mcmc.lq = apply(density.mcmc, 2, quantile, 0.025)
density.mcmc.uq = apply(density.mcmc, 2, quantile, 0.975)
plot(xx, density.mcmc.m, type="n",ylim=c(0,max(density.mcmc.uq)),xlab="Eruptions", ylab="Density")
polygon(c(xx,rev(xx)), c(density.mcmc.lq, rev(density.mcmc.uq)), col="grey", border="grey")
lines(xx, density.mcmc.m, col="black", lwd=2)
points(x, rep(0,n))
Provide an MCMC algorithm to fit the two-component, location-and-scale mixture of Gaussians.
### Loading data and setting up global variables
library(MASS)
library(MCMCpack)
data(faithful)
KK = 2 # As asked
x = faithful[,1]
n = length(x)
set.seed(781209)
### Get a "Bayesian" kernel density estimator based on the same location mixture of 2 normals
## Priors set up using an "empirical Bayes" approach
aa = rep(1,KK)
eta = mean(x)
tau = sqrt(var(x))
dd = 2
qq = var(x)/KK
w = rep(1,KK)/KK
mu = rnorm(KK, mean(x), sd(x))
sigma = rep(sd(x)/KK, KK)
cc = sample(1:KK, n, replace=T, prob=w)
## Number of iterations of the sampler
rrr = 12000
burn = 2000
## Storing the samples
cc.out = array(0, dim=c(rrr, n))
w.out = array(0, dim=c(rrr, KK))
mu.out = array(0, dim=c(rrr, KK))
sigma.out = array(0, dim=c(rrr, KK))
logpost = rep(0, rrr)
for(s in 1:rrr){
# Sample the indicators
for(i in 1:n){
v = rep(0,KK)
for(k in 1:KK){
v[k] = log(w[k]) + dnorm(x[i], mu[k], sigma[k], log=TRUE) #Compute the log of the weights
}
v = exp(v - max(v))/sum(exp(v - max(v)))
cc[i] = sample(1:KK, 1, replace=TRUE, prob=v)
}
# Sample the weights
w = as.vector(rdirichlet(1, aa + tabulate(cc, nbins=KK)))
# Sample the means
for(k in 1:KK){
nk = sum(cc==k)
xsumk = sum(x[cc==k])
tau2.hat = 1/(nk/sigma[k]^2 + 1/tau^2)
mu.hat = tau2.hat*(xsumk/sigma[k]^2 + eta/tau^2)
mu[k] = rnorm(1, mu.hat, sqrt(tau2.hat))
}
# Sample the variances
for(k in 1:KK) {
dd.star = dd + sum(cc==k)/2
sumsq = 0
for(i in 1:n) {
if (cc[i]==k) {
sumsq = sumsq + ((x[i]-mu[k])^2)
}
}
qq.star = qq + sumsq/2
sigma[k] = sqrt(rinvgamma(1, shape = dd.star, scale = qq.star))
}
# Store samples
cc.out[s,] = cc
w.out[s,] = w
mu.out[s,] = mu
sigma.out[s,] = sigma
for(i in 1:n){
logpost[s] = logpost[s] + log(w[cc[i]]) + dnorm(x[i], mu[cc[i]], sigma[cc[i]], log=TRUE)
}
logpost[s] = logpost[s] + log(ddirichlet(w, aa))
for(k in 1:KK){
logpost[s] = logpost[s] + dnorm(mu[k], eta, tau, log=TRUE)
logpost[s] = logpost[s] + log(dinvgamma(sigma[k]^2, shape = dd, scale = qq))
}
if(s/500==floor(s/500)){
print(paste("s =",s, w, mu, sigma, logpost[s]))
}
}
## [1] "s = 500 0.65972914904991 4.28469003268875 0.405452035141612 -284.691001291928" "s = 500 0.34027085095009 1.99006962679139 0.250995183146609 -284.691001291928"
## [1] "s = 1000 0.595942847037826 4.27800551958405 0.377348480705164 -286.969076980901" "s = 1000 0.404057152962174 2.03894151043637 0.30982681142742 -286.969076980901"
## [1] "s = 1500 0.647284731246712 4.32792289508716 0.42383087208717 -283.984683957035" "s = 1500 0.352715268753288 2.06264939688377 0.293003644635685 -283.984683957035"
## [1] "s = 2000 0.681875974292433 4.31964666156271 0.411719777867089 -284.392728508546" "s = 2000 0.318124025707567 2.05510718985465 0.295371255576304 -284.392728508546"
## [1] "s = 2500 0.601103182730055 4.24400160743064 0.446830400212041 -285.459906307434" "s = 2500 0.398896817269945 2.03587612253421 0.241473527538476 -285.459906307434"
## [1] "s = 3000 0.661235314614079 4.28417774604423 0.399650525878003 -285.992706414487" "s = 3000 0.338764685385921 2.07827156671072 0.297205073536538 -285.992706414487"
## [1] "s = 3500 0.644527213106916 4.28812055333749 0.441452138427281 -283.120726285281" "s = 3500 0.355472786893084 2.01146104149005 0.258057420679898 -283.120726285281"
## [1] "s = 4000 0.687810647492561 4.2844578377926 0.458980223432612 -287.498421765689" "s = 4000 0.312189352507439 2.07489936081685 0.259843371397385 -287.498421765689"
## [1] "s = 4500 0.675398367857913 4.23091315341658 0.410829138479542 -285.115602530381" "s = 4500 0.324601632142087 2.00548178399042 0.263106146522126 -285.115602530381"
## [1] "s = 5000 0.623677983258687 4.2986070783669 0.460990184040201 -287.679382635602" "s = 5000 0.376322016741313 2.04273694828362 0.296539486159871 -287.679382635602"
## [1] "s = 5500 0.650641729955276 4.30297831389609 0.410588171771523 -284.091323207027" "s = 5500 0.349358270044724 2.05603614578286 0.314091813936972 -284.091323207027"
## [1] "s = 6000 0.691002769510701 4.30695193759895 0.43643062048752 -285.25059244604" "s = 6000 0.308997230489299 2.00724366068866 0.258042136715897 -285.25059244604"
## [1] "s = 6500 0.639653531193993 4.30228067466188 0.466022578492273 -289.071680298941" "s = 6500 0.360346468806007 2.0370347870673 0.223458414339796 -289.071680298941"
## [1] "s = 7000 0.657266286407271 4.25015359390783 0.408697090783594 -285.609121794953" "s = 7000 0.342733713592729 1.98352245342408 0.281123375854885 -285.609121794953"
## [1] "s = 7500 0.582150245729404 4.27752884035096 0.405335835898845 -288.834567817749" "s = 7500 0.417849754270596 2.03280535197236 0.302512079952417 -288.834567817749"
## [1] "s = 8000 0.597745067402946 4.30797847743645 0.426676105386858 -286.886008869559" "s = 8000 0.402254932597054 1.97410330661178 0.266695259035665 -286.886008869559"
## [1] "s = 8500 0.67048977917052 4.34923962430531 0.440934482727733 -287.976277284725" "s = 8500 0.32951022082948 1.99450886009298 0.253385375719033 -287.976277284725"
## [1] "s = 9000 0.676526152024463 4.32975227643208 0.424164941801625 -286.420409681512" "s = 9000 0.323473847975537 2.03912276672083 0.304347017719118 -286.420409681512"
## [1] "s = 9500 0.675252958080913 4.32408885533291 0.42428978538769 -284.557683174045" "s = 9500 0.324747041919087 2.01338154879308 0.269521567887484 -284.557683174045"
## [1] "s = 10000 0.657398814367743 4.2933157987876 0.458707092652969 -283.811172985985" "s = 10000 0.342601185632257 2.03282618126385 0.248515730931108 -283.811172985985"
## [1] "s = 10500 0.662058917060533 4.31483111471995 0.427552521301718 -284.108178852953" "s = 10500 0.337941082939467 2.00950256527503 0.279659447389632 -284.108178852953"
## [1] "s = 11000 0.669552336247041 4.36928639963658 0.427369327130037 -288.518332324442" "s = 11000 0.330447663752959 1.97992561769347 0.296908732685882 -288.518332324442"
## [1] "s = 11500 0.627693164995977 4.20937955935309 0.437031029444333 -287.287695169602" "s = 11500 0.372306835004023 2.0025029877848 0.248798427816352 -287.287695169602"
## [1] "s = 12000 0.706166055902923 4.32329970006521 0.423366944818915 -291.385728914099" "s = 12000 0.293833944097077 2.09369936916931 0.272464988230699 -291.385728914099"
Provide code to generate the density estimate on a grid consisting of 150 points in the interval [0,7].
Eruptions = seq(0,7, length=150)
## Compute the samples of the density over a dense grid
density.mcmc = array(0, dim=c(rrr-burn,length(Eruptions)))
for(s in 1:(rrr-burn)){
for(k in 1:KK){
density.mcmc[s,] = density.mcmc[s,] + w.out[s+burn,k]*dnorm(Eruptions,mu.out[s+burn,k],sigma.out[s+burn,k])
}
}
density.mcmc.m = apply(density.mcmc , 2, mean)
## Plot Bayesian estimate with pointwise credible bands along with kernel density estimate and frequentist point estimate
colscale = c("black", "blue", "red")
#yy = density(x)
yy = density(Eruptions)
density.mcmc.lq = apply(density.mcmc, 2, quantile, 0.025)
density.mcmc.uq = apply(density.mcmc, 2, quantile, 0.975)
plot(Eruptions, density.mcmc.m, type="n",ylim=c(0,max(density.mcmc.uq)),xlab="Duration", ylab="Density")
polygon(c(Eruptions,rev(Eruptions)), c(density.mcmc.lq, rev(density.mcmc.uq)), col="grey", border="grey")
lines(Eruptions, density.mcmc.m, col=colscale[3], lwd=2)
#lines(xx, density.EM, col=colscale[2], lty=2, lwd=2)
#lines(yy, col=colscale[3], lty=3, lwd=2)
points(x, rep(0,n))
Provide the a graph of the density estimate on the interval [0,7].
Is the sampler for \(c\) correct?
The full conditional for \(c_i\) is given by,
\[ \Pr(c_i = 1|...) = \frac{\omega\frac{1}{\sqrt[]{2\pi\sigma_1}}exp\left\{-\frac{1}{2\sigma^2_1}(x_i-\mu_1)^2\right\}}{\omega\frac{1}{\sqrt[]{2\pi\sigma_1}}exp\left\{-\frac{1}{2\sigma^2_1}(x_i-\mu_1)^2\right\}+(1-\omega)\frac{1}{\sqrt[]{2\pi\sigma_2}}exp\left\{-\frac{1}{2\sigma^2_2}(x_i-\mu_2)^2\right\}} \]
(This assumes that the mixture weights are parameterized as
for(i in 1:n){
v = rep(0,2)
v[1] = log(w) + dnorm(x[i], mu[1], sigma[1], log=TRUE) #Compute the log of the weights
v[2] = log(1-w) + dnorm(x[i], mu[2], sigma[2], log=TRUE) #Compute the log of the weights
v = exp(v - max(v))/sum(exp(v - max(v)))
cc[i] = sample(1:2, 1, replace=TRUE, prob=v)
}
Is the sampler for \(\omega\) correct?
The full conditional is simply
\[ \omega \mid \cdots \mbox{Beta} \left(\alpha_1 + \sum_{i=1}^{n} \mathbf{1}(c_i=1), \alpha_2 + \sum_{i=1}^{n} \mathbf{1}(c_i=2)\right) \]
The following one line of code implements it:
w = rbeta(1, aa[1] + sum(cc==1), aa[2] + sum(cc==2))
Are the sampler for \(\mu\) correct?
\[ \mu_k \mid \cdots \mbox{Normal}\left( \left[ \frac{n_k}{\sigma_k^2} + \frac{1}{\tau^2} \right]^{-1}\left[ \frac{\sum_{i:c_i=k} x_i}{\sigma_k^2} + \frac{\eta}{\tau^2} \right], \left[ \frac{n_k}{\sigma_k^2} + \frac{1}{\tau^2} \right]^{-1}\right) \]
The following code implements the sampler:
# Sample the means
for(k in 1:2){
nk = sum(cc==k)
xsumk = sum(x[cc==k])
tau2.hat = 1/(nk/sigma[k]^2 + 1/tau^2)
mu.hat = tau2.hat*(xsumk/sigma[k]^2 + eta/tau^2)
mu[k] = rnorm(1, mu.hat, sqrt(tau2.hat))
}
Remember, however, that there are various ways in which the implementation could proceed.
Are the samplers for \(\sigma_k\) correct?
The full conditional takes the form:
\[ 1/\sigma^2_k \mid \cdots \mbox{Gamma}\left(d + \frac{n_k}{2} , q + \frac{1}{2} \sum_{i:c_k=k} (x_i - \mu_k)^2 \right) \]
The following code implements the sampler:
# Sample the variances
for(k in 1:2){
nk = sum(cc==k)
dd.star = dd + nk/2
qq.star = qq + sum((x[cc==k] - mu[k])^2)/2
sigma[k] = sqrt(1/rgamma(1, dd.star, qq.star))
}
Remember, however, that there are various ways in which the implementation could proceed.
Does the code to generate the density estimate appear correct?
This is an example that, as before, assumes that the weights are parameterized as \(\omega\) and \(1- \omega\).
## Plot Bayesian estimate with pointwise credible bands
density.mcmc.lq = apply(density.mcmc, 2, quantile, 0.025)
density.mcmc.uq = apply(density.mcmc, 2, quantile, 0.975)
plot(xx, density.mcmc.m, type="n",ylim=c(0,max(density.mcmc.uq)),xlab="Eruptions", ylab="Density")
polygon(c(xx,rev(xx)), c(density.mcmc.lq, rev(density.mcmc.uq)), col="grey", border="grey")
lines(xx, density.mcmc.m, col="black", lwd=2)
points(x, rep(0,n))
Note that this will generate not only a point estimate, but also pointwise credible intervals.
Does the graph appear to be correct?
The density estimate should look similar to the Figure below. Note that the variance of both components is clearly different.
The full code for the MCMC algorithm follows:
x = faithful$eruptions
n = length(x)
plot(density(x))
points(x,rep(0,n))
## Priors set up using an "empirical Bayes" approach
aa = rep(1,2)
eta = mean(x)
tau = sqrt(var(x))
dd = 2
qq = var(x)/2
## Initialize the parameters
w = 1/2
mu = rnorm(2, mean(x), sd(x))
sigma = rep(sd(x)/2,2)
cc = sample(1:2, n, replace=T, prob=c(1/2,1/2))
## Number of iterations of the sampler
rrr = 12000
burn = 2000
## Storing the samples
cc.out = array(0, dim=c(rrr, n))
w.out = rep(0, rrr)
mu.out = array(0, dim=c(rrr, 2))
sigma.out = array(0, dim=c(rrr, 2))
logpost = rep(0, rrr)
for(s in 1:rrr){
# Sample the indicators
for(i in 1:n){
v = rep(0,2)
v[1] = log(w) + dnorm(x[i], mu[1], sigma[1], log=TRUE) #Compute the log of the weights
v[2] = log(1-w) + dnorm(x[i], mu[2], sigma[2], log=TRUE) #Compute the log of the weights
v = exp(v - max(v))/sum(exp(v - max(v)))
cc[i] = sample(1:2, 1, replace=TRUE, prob=v)
}
# Sample the weights
w = rbeta(1, aa[1] + sum(cc==1), aa[2] + sum(cc==2))
# Sample the parameters of the components
for(k in 1:2){
# Sample the means
nk = sum(cc==k)
xsumk = sum(x[cc==k])
tau2.hat = 1/(nk/sigma[k]^2 + 1/tau^2)
mu.hat = tau2.hat*(xsumk/sigma[k]^2 + eta/tau^2)
mu[k] = rnorm(1, mu.hat, sqrt(tau2.hat))
# Sample the variances
dd.star = dd + nk/2
qq.star = qq + sum((x[cc==k] - mu[k])^2)/2
sigma[k] = sqrt(1/rgamma(1, dd.star, qq.star))
}
# Store samples
cc.out[s,] = cc
w.out[s] = w
mu.out[s,] = mu
sigma.out[s,] = sigma
logpost[s] = 0
for(i in 1:n){
if(cc[i] == 1){
logpost[s] = logpost[s] + log(w) + dnorm(x[i], mu[1], sigma[1], log=TRUE)
}else{
logpost[s] = logpost[s] + log(1-w) + dnorm(x[i], mu[2], sigma[2], log=TRUE)
}
}
logpost[s] = logpost[s] + dbeta(w, aa[1], aa[2], log=TRUE)
for(k in 1:2){
logpost[s] = logpost[s] + dnorm(mu[k], eta, tau, log = T) + dgamma(1/sigma[k]^2, dd, qq)/sigma[k]^4
}
if(s/500==floor(s/500)){
print(paste("s =",s))
}
}
## [1] "s = 500"
## [1] "s = 1000"
## [1] "s = 1500"
## [1] "s = 2000"
## [1] "s = 2500"
## [1] "s = 3000"
## [1] "s = 3500"
## [1] "s = 4000"
## [1] "s = 4500"
## [1] "s = 5000"
## [1] "s = 5500"
## [1] "s = 6000"
## [1] "s = 6500"
## [1] "s = 7000"
## [1] "s = 7500"
## [1] "s = 8000"
## [1] "s = 8500"
## [1] "s = 9000"
## [1] "s = 9500"
## [1] "s = 10000"
## [1] "s = 10500"
## [1] "s = 11000"
## [1] "s = 11500"
## [1] "s = 12000"
## Compute the samples of the density over a dense grid
xx = seq(0,7,length=150)
density.mcmc = array(0, dim=c(rrr-burn,length(xx)))
for(s in 1:(rrr-burn)){
density.mcmc[s,] = density.mcmc[s,] + w.out[s+burn]*dnorm(xx,mu.out[s+burn,1],sigma.out[s+burn,1]) +
(1-w.out[s+burn])*dnorm(xx,mu.out[s+burn,2],sigma.out[s+burn,2])
}
density.mcmc.m = apply(density.mcmc , 2, mean)
## Plot Bayesian estimate with pointwise credible bands
density.mcmc.lq = apply(density.mcmc, 2, quantile, 0.025)
density.mcmc.uq = apply(density.mcmc, 2, quantile, 0.975)
plot(xx, density.mcmc.m, type="n",ylim=c(0,max(density.mcmc.uq)),xlab="Eruptions", ylab="Density")
polygon(c(xx,rev(xx)), c(density.mcmc.lq, rev(density.mcmc.uq)), col="grey", border="grey")
lines(xx, density.mcmc.m, col="black", lwd=2)
points(x, rep(0,n))
Provide an MCMC algorithm to fit the two-component, location-and-scale mixture of Gaussians.
### Get a "Bayesian" kernel density estimator based on the same location mixture of 6 normals
## Priors set up using an "empirical Bayes" approach
aa = rep(1,KK)
eta = mean(x)
tau = sqrt(var(x))
dd = 2
qq = var(x)/KK
mu_0 = rnorm(KK, eta, tau)
sigma_0 = rinvgamma(KK, dd, qq)
## Initialize the parameters
w = rep(1,KK)/KK
mu = rnorm(KK, mean(x), sd(x))
sigma = sd(x)/KK
cc = sample(1:KK, n, replace=T, prob=w)
## Number of iterations of the sampler
rrr = 12000
burn = 2000
## Storing the samples
cc.out = array(0, dim=c(rrr, n))
w.out = array(0, dim=c(rrr, KK))
mu.out = array(0, dim=c(rrr, KK))
sigma.out = rep(0, rrr)
logpost = rep(0, rrr)
for(s in 1:rrr){
# Sample the indicators
for(i in 1:n){
v = rep(0,KK)
for(k in 1:KK){
v[k] = log(w[k]) + dnorm(x[i], mu[k], sigma, log=TRUE) #Compute the log of the weights
}
v = exp(v - max(v))/sum(exp(v - max(v)))
cc[i] = sample(1:KK, 1, replace=TRUE, prob=v)
}
# Sample the weights
## https://cran.r-project.org/web/packages/rBeta2009/rBeta2009.pdf
w = as.vector(rdirichlet(1, aa + tabulate(cc, nbins=KK)))
# Sample the means
for(k in 1:KK){
nk = sum(cc==k)
xsumk = sum(x[cc==k])
tau2.hat = 1/(nk/sigma^2 + 1/sigma_0[k]^2)
mu.hat = tau2.hat*(xsumk/sigma^2 + mu_0[k]/sigma_0[k]^2)
mu[k] = rnorm(1, mu.hat, sqrt(tau2.hat))
}
# Sample the variances
dd.star = dd + n/2
qq.star = qq + sum((x - mu[cc])^2)/2
sigma = sqrt(1/rgamma(1, dd.star, qq.star))
# Store samples
cc.out[s,] = cc
w.out[s,] = w
mu.out[s,] = mu
sigma.out[s] = sigma
for(i in 1:n){
logpost[s] = logpost[s] + log(w[cc[i]]) + dnorm(x[i], mu[cc[i]], sigma, log=TRUE)
}
logpost[s] = logpost[s] + log(ddirichlet(w, aa))
for(k in 1:KK){
logpost[s] = logpost[s] + dnorm(mu[k], eta, tau, log=TRUE)
}
logpost[s] = logpost[s] + dgamma(1/sigma^2, dd, qq, log=TRUE) - 4*log(sigma)
if(s/500==floor(s/500)){
print(paste("s =",s))
}
}
## [1] "s = 500"
## [1] "s = 1000"
## [1] "s = 1500"
## [1] "s = 2000"
## [1] "s = 2500"
## [1] "s = 3000"
## [1] "s = 3500"
## [1] "s = 4000"
## [1] "s = 4500"
## [1] "s = 5000"
## [1] "s = 5500"
## [1] "s = 6000"
## [1] "s = 6500"
## [1] "s = 7000"
## [1] "s = 7500"
## [1] "s = 8000"
## [1] "s = 8500"
## [1] "s = 9000"
## [1] "s = 9500"
## [1] "s = 10000"
## [1] "s = 10500"
## [1] "s = 11000"
## [1] "s = 11500"
## [1] "s = 12000"
Provide code to generate the density estimate on a grid consisting of 150 points in the interval [0,7].
xx = seq(0,7,length=150)
nxx = length(xx)
## Compute the samples of the density over a dense grid
density.mcmc = array(0, dim=c(rrr-burn,length(xx)))
for(s in 1:(rrr-burn)){
for(k in 1:KK){
density.mcmc[s,] = density.mcmc[s,] + w.out[s+burn,k]*dnorm(xx,mu.out[s+burn,k],sigma.out[s+burn])
}
}
density.mcmc.m = apply(density.mcmc , 2, mean)
Provide the a graph of the density estimate on the interval [0,7].
## Plot Bayesian estimate with pointwise credible bands along with kernel density estimate and frequentist point estimate
colscale = c("black", "blue", "red")
yy = density(x)
density.mcmc.lq = apply(density.mcmc, 2, quantile, 0.025)
density.mcmc.uq = apply(density.mcmc, 2, quantile, 0.975)
plot(xx, density.mcmc.m, type="n",ylim=c(0,max(density.mcmc.uq)),xlab="Eruption Duration", ylab="Density")
polygon(c(xx,rev(xx)), c(density.mcmc.lq, rev(density.mcmc.uq)), col="grey", border="grey")
lines(xx, density.mcmc.m, col=colscale[1], lwd=2)
lines(xx, density.EM, col=colscale[2], lty=2, lwd=2)
## Error in xy.coords(x, y): object 'density.EM' not found
lines(yy, col=colscale[3], lty=3, lwd=2)
points(x, rep(0,n))
legend(5, 0.45, c("KDE","EM","MCMC"), col=colscale[c(3,2,1)], lty=c(3,2,1), lwd=2, bty="n")
Is the sampler for \(c\) correct?
The full conditional for \(c_i\) is given by,
\[ \Pr(c_i = 1|...) = \frac{\omega\frac{1}{\sqrt[]{2\pi\sigma_1}}exp\left\{-\frac{1}{2\sigma^2_1}(x_i-\mu_1)^2\right\}}{\omega\frac{1}{\sqrt[]{2\pi\sigma_1}}exp\left\{-\frac{1}{2\sigma^2_1}(x_i-\mu_1)^2\right\}+(1-\omega)\frac{1}{\sqrt[]{2\pi\sigma_2}}exp\left\{-\frac{1}{2\sigma^2_2}(x_i-\mu_2)^2\right\}} \]
(This assumes that the mixture weights are parameterized as
for(i in 1:n){
v = rep(0,2)
v[1] = log(w) + dnorm(x[i], mu[1], sigma[1], log=TRUE) #Compute the log of the weights
v[2] = log(1-w) + dnorm(x[i], mu[2], sigma[2], log=TRUE) #Compute the log of the weights
v = exp(v - max(v))/sum(exp(v - max(v)))
cc[i] = sample(1:2, 1, replace=TRUE, prob=v)
}
## Error in sample.int(length(x), size, replace, prob): NA in probability vector
Is the sampler for \(\omega\) correct?
The full conditional is simply
\[ \omega \mid \cdots \mbox{Beta} \left(\alpha_1 + \sum_{i=1}^{n} \mathbf{1}(c_i=1), \alpha_2 + \sum_{i=1}^{n} \mathbf{1}(c_i=2)\right) \]
The following one line of code implements it:
w = rbeta(1, aa[1] + sum(cc==1), aa[2] + sum(cc==2))
Are the sampler for \(\mu\) correct?
\[ \mu_k \mid \cdots \mbox{Normal}\left( \left[ \frac{n_k}{\sigma_k^2} + \frac{1}{\tau^2} \right]^{-1}\left[ \frac{\sum_{i:c_i=k} x_i}{\sigma_k^2} + \frac{\eta}{\tau^2} \right], \left[ \frac{n_k}{\sigma_k^2} + \frac{1}{\tau^2} \right]^{-1}\right) \]
The following code implements the sampler:
# Sample the means
for(k in 1:2){
nk = sum(cc==k)
xsumk = sum(x[cc==k])
tau2.hat = 1/(nk/sigma[k]^2 + 1/tau^2)
mu.hat = tau2.hat*(xsumk/sigma[k]^2 + eta/tau^2)
mu[k] = rnorm(1, mu.hat, sqrt(tau2.hat))
}
Remember, however, that there are various ways in which the implementation could proceed.
Are the samplers for \(\sigma_k\) correct?
The full conditional takes the form:
\[ 1/\sigma^2_k \mid \cdots \mbox{Gamma}\left(d + \frac{n_k}{2} , q + \frac{1}{2} \sum_{i:c_k=k} (x_i - \mu_k)^2 \right) \]
The following code implements the sampler:
# Sample the variances
for(k in 1:2){
nk = sum(cc==k)
dd.star = dd + nk/2
qq.star = qq + sum((x[cc==k] - mu[k])^2)/2
sigma[k] = sqrt(1/rgamma(1, dd.star, qq.star))
}
Remember, however, that there are various ways in which the implementation could proceed.
Does the code to generate the density estimate appear correct?
This is an example that, as before, assumes that the weights are parameterized as \(\omega\) and \(1- \omega\).
## Plot Bayesian estimate with pointwise credible bands
density.mcmc.lq = apply(density.mcmc, 2, quantile, 0.025)
density.mcmc.uq = apply(density.mcmc, 2, quantile, 0.975)
plot(xx, density.mcmc.m, type="n",ylim=c(0,max(density.mcmc.uq)),xlab="Eruptions", ylab="Density")
polygon(c(xx,rev(xx)), c(density.mcmc.lq, rev(density.mcmc.uq)), col="grey", border="grey")
lines(xx, density.mcmc.m, col="black", lwd=2)
points(x, rep(0,n))
Note that this will generate not only a point estimate, but also pointwise credible intervals.
Does the graph appear to be correct?
The density estimate should look similar to the Figure below. Note that the variance of both components is clearly different.
The full code for the MCMC algorithm follows:
x = faithful$eruptions
n = length(x)
plot(density(x))
points(x,rep(0,n))
## Priors set up using an "empirical Bayes" approach
aa = rep(1,2)
eta = mean(x)
tau = sqrt(var(x))
dd = 2
qq = var(x)/2
## Initialize the parameters
w = 1/2
mu = rnorm(2, mean(x), sd(x))
sigma = rep(sd(x)/2,2)
cc = sample(1:2, n, replace=T, prob=c(1/2,1/2))
## Number of iterations of the sampler
rrr = 12000
burn = 2000
## Storing the samples
cc.out = array(0, dim=c(rrr, n))
w.out = rep(0, rrr)
mu.out = array(0, dim=c(rrr, 2))
sigma.out = array(0, dim=c(rrr, 2))
logpost = rep(0, rrr)
for(s in 1:rrr){
# Sample the indicators
for(i in 1:n){
v = rep(0,2)
v[1] = log(w) + dnorm(x[i], mu[1], sigma[1], log=TRUE) #Compute the log of the weights
v[2] = log(1-w) + dnorm(x[i], mu[2], sigma[2], log=TRUE) #Compute the log of the weights
v = exp(v - max(v))/sum(exp(v - max(v)))
cc[i] = sample(1:2, 1, replace=TRUE, prob=v)
}
# Sample the weights
w = rbeta(1, aa[1] + sum(cc==1), aa[2] + sum(cc==2))
# Sample the parameters of the components
for(k in 1:2){
# Sample the means
nk = sum(cc==k)
xsumk = sum(x[cc==k])
tau2.hat = 1/(nk/sigma[k]^2 + 1/tau^2)
mu.hat = tau2.hat*(xsumk/sigma[k]^2 + eta/tau^2)
mu[k] = rnorm(1, mu.hat, sqrt(tau2.hat))
# Sample the variances
dd.star = dd + nk/2
qq.star = qq + sum((x[cc==k] - mu[k])^2)/2
sigma[k] = sqrt(1/rgamma(1, dd.star, qq.star))
}
# Store samples
cc.out[s,] = cc
w.out[s] = w
mu.out[s,] = mu
sigma.out[s,] = sigma
logpost[s] = 0
for(i in 1:n){
if(cc[i] == 1){
logpost[s] = logpost[s] + log(w) + dnorm(x[i], mu[1], sigma[1], log=TRUE)
}else{
logpost[s] = logpost[s] + log(1-w) + dnorm(x[i], mu[2], sigma[2], log=TRUE)
}
}
logpost[s] = logpost[s] + dbeta(w, aa[1], aa[2], log=TRUE)
for(k in 1:2){
logpost[s] = logpost[s] + dnorm(mu[k], eta, tau, log = T) + dgamma(1/sigma[k]^2, dd, qq)/sigma[k]^4
}
if(s/500==floor(s/500)){
print(paste("s =",s))
}
}
## [1] "s = 500"
## [1] "s = 1000"
## [1] "s = 1500"
## [1] "s = 2000"
## [1] "s = 2500"
## [1] "s = 3000"
## [1] "s = 3500"
## [1] "s = 4000"
## [1] "s = 4500"
## [1] "s = 5000"
## [1] "s = 5500"
## [1] "s = 6000"
## [1] "s = 6500"
## [1] "s = 7000"
## [1] "s = 7500"
## [1] "s = 8000"
## [1] "s = 8500"
## [1] "s = 9000"
## [1] "s = 9500"
## [1] "s = 10000"
## [1] "s = 10500"
## [1] "s = 11000"
## [1] "s = 11500"
## [1] "s = 12000"
## Compute the samples of the density over a dense grid
xx = seq(0,7,length=150)
density.mcmc = array(0, dim=c(rrr-burn,length(xx)))
for(s in 1:(rrr-burn)){
density.mcmc[s,] = density.mcmc[s,] + w.out[s+burn]*dnorm(xx,mu.out[s+burn,1],sigma.out[s+burn,1]) +
(1-w.out[s+burn])*dnorm(xx,mu.out[s+burn,2],sigma.out[s+burn,2])
}
density.mcmc.m = apply(density.mcmc , 2, mean)
## Plot Bayesian estimate with pointwise credible bands
density.mcmc.lq = apply(density.mcmc, 2, quantile, 0.025)
density.mcmc.uq = apply(density.mcmc, 2, quantile, 0.975)
plot(xx, density.mcmc.m, type="n",ylim=c(0,max(density.mcmc.uq)),xlab="Eruptions", ylab="Density")
polygon(c(xx,rev(xx)), c(density.mcmc.lq, rev(density.mcmc.uq)), col="grey", border="grey")
lines(xx, density.mcmc.m, col="black", lwd=2)
points(x, rep(0,n))
Provide an MCMC algorithm to fit the two-component, location-and-scale mixture of Gaussians.
### Loading data and setting up global variables
library(MASS)
library(MCMCpack)
data(faithful)
KK = 2 # As asked
x = faithful[,1]
n = length(x)
set.seed(781209)
### Get a "Bayesian" kernel density estimator based on the same location mixture of 2 normals
## Priors set up using an "empirical Bayes" approach
aa = rep(1,KK)
eta = mean(x)
tau = sqrt(var(x))
dd = 2
qq = var(x)/KK
w = rep(1,KK)/KK
mu = rnorm(KK, mean(x), sd(x))
sigma = rep(sd(x)/KK, KK)
cc = sample(1:KK, n, replace=T, prob=w)
## Number of iterations of the sampler
rrr = 12000
burn = 2000
## Storing the samples
cc.out = array(0, dim=c(rrr, n))
w.out = array(0, dim=c(rrr, KK))
mu.out = array(0, dim=c(rrr, KK))
sigma.out = array(0, dim=c(rrr, KK))
logpost = rep(0, rrr)
for(s in 1:rrr){
# Sample the indicators
for(i in 1:n){
v = rep(0,KK)
for(k in 1:KK){
v[k] = log(w[k]) + dnorm(x[i], mu[k], sigma[k], log=TRUE) #Compute the log of the weights
}
v = exp(v - max(v))/sum(exp(v - max(v)))
cc[i] = sample(1:KK, 1, replace=TRUE, prob=v)
}
# Sample the weights
w = as.vector(rdirichlet(1, aa + tabulate(cc, nbins=KK)))
# Sample the means
for(k in 1:KK){
nk = sum(cc==k)
xsumk = sum(x[cc==k])
tau2.hat = 1/(nk/sigma[k]^2 + 1/tau^2)
mu.hat = tau2.hat*(xsumk/sigma[k]^2 + eta/tau^2)
mu[k] = rnorm(1, mu.hat, sqrt(tau2.hat))
}
# Sample the variances
for(k in 1:KK) {
dd.star = dd + sum(cc==k)/2
sumsq = 0
for(i in 1:n) {
if (cc[i]==k) {
sumsq = sumsq + ((x[i]-mu[k])^2)
}
}
qq.star = qq + sumsq/2
sigma[k] = sqrt(MCMCpack::rinvgamma(1, shape = dd.star, scale = qq.star))
}
# Store samples
cc.out[s,] = cc
w.out[s,] = w
mu.out[s,] = mu
sigma.out[s,] = sigma
for(i in 1:n){
logpost[s] = logpost[s] + log(w[cc[i]]) + dnorm(x[i], mu[cc[i]], sigma[cc[i]], log=TRUE)
}
logpost[s] = logpost[s] + log(ddirichlet(w, aa))
for(k in 1:KK){
logpost[s] = logpost[s] + dnorm(mu[k], eta, tau, log=TRUE)
logpost[s] = logpost[s] + log(MCMCpack::dinvgamma(sigma[k]^2, shape = dd, scale = qq))
}
if(s/500==floor(s/500)){
print(paste("s =",s, w, mu, sigma, logpost[s]))
}
}
## [1] "s = 500 0.65972914904991 4.28469003268875 0.405452035141612 -284.691001291928" "s = 500 0.34027085095009 1.99006962679139 0.250995183146609 -284.691001291928"
## [1] "s = 1000 0.595942847037826 4.27800551958405 0.377348480705164 -286.969076980901" "s = 1000 0.404057152962174 2.03894151043637 0.30982681142742 -286.969076980901"
## [1] "s = 1500 0.647284731246712 4.32792289508716 0.42383087208717 -283.984683957035" "s = 1500 0.352715268753288 2.06264939688377 0.293003644635685 -283.984683957035"
## [1] "s = 2000 0.681875974292433 4.31964666156271 0.411719777867089 -284.392728508546" "s = 2000 0.318124025707567 2.05510718985465 0.295371255576304 -284.392728508546"
## [1] "s = 2500 0.601103182730055 4.24400160743064 0.446830400212041 -285.459906307434" "s = 2500 0.398896817269945 2.03587612253421 0.241473527538476 -285.459906307434"
## [1] "s = 3000 0.661235314614079 4.28417774604423 0.399650525878003 -285.992706414487" "s = 3000 0.338764685385921 2.07827156671072 0.297205073536538 -285.992706414487"
## [1] "s = 3500 0.644527213106916 4.28812055333749 0.441452138427281 -283.120726285281" "s = 3500 0.355472786893084 2.01146104149005 0.258057420679898 -283.120726285281"
## [1] "s = 4000 0.687810647492561 4.2844578377926 0.458980223432612 -287.498421765689" "s = 4000 0.312189352507439 2.07489936081685 0.259843371397385 -287.498421765689"
## [1] "s = 4500 0.675398367857913 4.23091315341658 0.410829138479542 -285.115602530381" "s = 4500 0.324601632142087 2.00548178399042 0.263106146522126 -285.115602530381"
## [1] "s = 5000 0.623677983258687 4.2986070783669 0.460990184040201 -287.679382635602" "s = 5000 0.376322016741313 2.04273694828362 0.296539486159871 -287.679382635602"
## [1] "s = 5500 0.650641729955276 4.30297831389609 0.410588171771523 -284.091323207027" "s = 5500 0.349358270044724 2.05603614578286 0.314091813936972 -284.091323207027"
## [1] "s = 6000 0.691002769510701 4.30695193759895 0.43643062048752 -285.25059244604" "s = 6000 0.308997230489299 2.00724366068866 0.258042136715897 -285.25059244604"
## [1] "s = 6500 0.639653531193993 4.30228067466188 0.466022578492273 -289.071680298941" "s = 6500 0.360346468806007 2.0370347870673 0.223458414339796 -289.071680298941"
## [1] "s = 7000 0.657266286407271 4.25015359390783 0.408697090783594 -285.609121794953" "s = 7000 0.342733713592729 1.98352245342408 0.281123375854885 -285.609121794953"
## [1] "s = 7500 0.582150245729404 4.27752884035096 0.405335835898845 -288.834567817749" "s = 7500 0.417849754270596 2.03280535197236 0.302512079952417 -288.834567817749"
## [1] "s = 8000 0.597745067402946 4.30797847743645 0.426676105386858 -286.886008869559" "s = 8000 0.402254932597054 1.97410330661178 0.266695259035665 -286.886008869559"
## [1] "s = 8500 0.67048977917052 4.34923962430531 0.440934482727733 -287.976277284725" "s = 8500 0.32951022082948 1.99450886009298 0.253385375719033 -287.976277284725"
## [1] "s = 9000 0.676526152024463 4.32975227643208 0.424164941801625 -286.420409681512" "s = 9000 0.323473847975537 2.03912276672083 0.304347017719118 -286.420409681512"
## [1] "s = 9500 0.675252958080913 4.32408885533291 0.42428978538769 -284.557683174045" "s = 9500 0.324747041919087 2.01338154879308 0.269521567887484 -284.557683174045"
## [1] "s = 10000 0.657398814367743 4.2933157987876 0.458707092652969 -283.811172985985" "s = 10000 0.342601185632257 2.03282618126385 0.248515730931108 -283.811172985985"
## [1] "s = 10500 0.662058917060533 4.31483111471995 0.427552521301718 -284.108178852953" "s = 10500 0.337941082939467 2.00950256527503 0.279659447389632 -284.108178852953"
## [1] "s = 11000 0.669552336247041 4.36928639963658 0.427369327130037 -288.518332324442" "s = 11000 0.330447663752959 1.97992561769347 0.296908732685882 -288.518332324442"
## [1] "s = 11500 0.627693164995977 4.20937955935309 0.437031029444333 -287.287695169602" "s = 11500 0.372306835004023 2.0025029877848 0.248798427816352 -287.287695169602"
## [1] "s = 12000 0.706166055902923 4.32329970006521 0.423366944818915 -291.385728914099" "s = 12000 0.293833944097077 2.09369936916931 0.272464988230699 -291.385728914099"
Provide code to generate the density estimate on a grid consisting of 150 points in the interval [0,7].
Eruptions = seq(0,7, length.out=150)
## Compute the samples of the density over a dense grid
density.mcmc = array(0, dim=c(rrr-burn,length(Eruptions)))
for(s in 1:(rrr-burn)){
for(k in 1:KK){
density.mcmc[s,] = density.mcmc[s,] + w.out[s+burn,k]*dnorm(Eruptions,mu.out[s+burn,k],sigma.out[s+burn,k])
}
}
density.mcmc.m = apply(density.mcmc , 2, mean)
## Plot Bayesian estimate with pointwise credible bands along with kernel density estimate and frequentist point estimate
colscale = c("black", "blue", "red")
#yy = density(x)
yy = density(Eruptions)
density.mcmc.lq = replace_na(apply(density.mcmc, 2, quantile, 0.025, na.rm=TRUE), 0)
density.mcmc.uq = replace_na(apply(density.mcmc, 2, quantile, 0.975, na.rm=TRUE), 0)
plot(Eruptions, density.mcmc.m, type="n", ylim=c(0,max(density.mcmc.uq)), xlab="Duration", ylab="Density")
polygon(c(Eruptions,rev(Eruptions)), c(density.mcmc.lq, rev(density.mcmc.uq)), col="grey", border="grey")
lines(Eruptions, density.mcmc.m, col=colscale[3], lwd=2)
#lines(xx, density.EM, col=colscale[2], lty=2, lwd=2)
#lines(yy, col=colscale[3], lty=3, lwd=2)
points(x, rep(0,n))
Provide the a graph of the density estimate on the interval [0,7].
Is the sampler for \(c\) correct?
The full conditional for \(c_i\) is given by,
\[ \Pr(c_i = 1|...) = \frac{\omega\frac{1}{\sqrt[]{2\pi\sigma_1}}exp\left\{-\frac{1}{2\sigma^2_1}(x_i-\mu_1)^2\right\}}{\omega\frac{1}{\sqrt[]{2\pi\sigma_1}}exp\left\{-\frac{1}{2\sigma^2_1}(x_i-\mu_1)^2\right\}+(1-\omega)\frac{1}{\sqrt[]{2\pi\sigma_2}}exp\left\{-\frac{1}{2\sigma^2_2}(x_i-\mu_2)^2\right\}} \]
(This assumes that the mixture weights are parameterized as
for(i in 1:n){
v = rep(0,2)
v[1] = log(w) + dnorm(x[i], mu[1], sigma[1], log=TRUE) #Compute the log of the weights
v[2] = log(1-w) + dnorm(x[i], mu[2], sigma[2], log=TRUE) #Compute the log of the weights
v = exp(v - max(v))/sum(exp(v - max(v)))
cc[i] = sample(1:2, 1, replace=TRUE, prob=v)
}
Is the sampler for \(\omega\) correct?
The full conditional is simply
\[ \omega \mid \cdots \mbox{Beta} \left(\alpha_1 + \sum_{i=1}^{n} \mathbf{1}(c_i=1), \alpha_2 + \sum_{i=1}^{n} \mathbf{1}(c_i=2)\right) \]
The following one line of code implements it:
w = rbeta(1, aa[1] + sum(cc==1), aa[2] + sum(cc==2))
Are the sampler for \(\mu\) correct?
\[ \mu_k \mid \cdots \mbox{Normal}\left( \left[ \frac{n_k}{\sigma_k^2} + \frac{1}{\tau^2} \right]^{-1}\left[ \frac{\sum_{i:c_i=k} x_i}{\sigma_k^2} + \frac{\eta}{\tau^2} \right], \left[ \frac{n_k}{\sigma_k^2} + \frac{1}{\tau^2} \right]^{-1}\right) \]
The following code implements the sampler:
# Sample the means
for(k in 1:2){
nk = sum(cc==k)
xsumk = sum(x[cc==k])
tau2.hat = 1/(nk/sigma[k]^2 + 1/tau^2)
mu.hat = tau2.hat*(xsumk/sigma[k]^2 + eta/tau^2)
mu[k] = rnorm(1, mu.hat, sqrt(tau2.hat))
}
Remember, however, that there are various ways in which the implementation could proceed.
Are the samplers for \(\sigma_k\) correct?
The full conditional takes the form:
\[ 1/\sigma^2_k \mid \cdots \mbox{Gamma}\left(d + \frac{n_k}{2} , q + \frac{1}{2} \sum_{i:c_k=k} (x_i - \mu_k)^2 \right) \]
The following code implements the sampler:
# Sample the variances
for(k in 1:2){
nk = sum(cc==k)
dd.star = dd + nk/2
qq.star = qq + sum((x[cc==k] - mu[k])^2)/2
sigma[k] = sqrt(1/rgamma(1, dd.star, qq.star))
}
Remember, however, that there are various ways in which the implementation could proceed.
Does the code to generate the density estimate appear correct?
This is an example that, as before, assumes that the weights are parameterized as \(\omega\) and \(1- \omega\).
## Plot Bayesian estimate with pointwise credible bands
density.mcmc.lq = apply(density.mcmc, 2, quantile, 0.025)
density.mcmc.uq = apply(density.mcmc, 2, quantile, 0.975)
plot(xx, density.mcmc.m, type="n",ylim=c(0,max(density.mcmc.uq)),xlab="Eruptions", ylab="Density")
polygon(c(xx,rev(xx)), c(density.mcmc.lq, rev(density.mcmc.uq)), col="grey", border="grey")
lines(xx, density.mcmc.m, col="black", lwd=2)
points(x, rep(0,n))
Note that this will generate not only a point estimate, but also pointwise credible intervals.
Does the graph appear to be correct?
The density estimate should look similar to the Figure below. Note that the variance of both components is clearly different.
The full code for the MCMC algorithm follows:
x = faithful$eruptions
n = length(x)
plot(density(x))
points(x,rep(0,n))
## Priors set up using an "empirical Bayes" approach
aa = rep(1,2)
eta = mean(x)
tau = sqrt(var(x))
dd = 2
qq = var(x)/2
## Initialize the parameters
w = 1/2
mu = rnorm(2, mean(x), sd(x))
sigma = rep(sd(x)/2,2)
cc = sample(1:2, n, replace=T, prob=c(1/2,1/2))
## Number of iterations of the sampler
rrr = 12000
burn = 2000
## Storing the samples
cc.out = array(0, dim=c(rrr, n))
w.out = rep(0, rrr)
mu.out = array(0, dim=c(rrr, 2))
sigma.out = array(0, dim=c(rrr, 2))
logpost = rep(0, rrr)
for(s in 1:rrr){
# Sample the indicators
for(i in 1:n){
v = rep(0,2)
v[1] = log(w) + dnorm(x[i], mu[1], sigma[1], log=TRUE) #Compute the log of the weights
v[2] = log(1-w) + dnorm(x[i], mu[2], sigma[2], log=TRUE) #Compute the log of the weights
v = exp(v - max(v))/sum(exp(v - max(v)))
cc[i] = sample(1:2, 1, replace=TRUE, prob=v)
}
# Sample the weights
w = rbeta(1, aa[1] + sum(cc==1), aa[2] + sum(cc==2))
# Sample the parameters of the components
for(k in 1:2){
# Sample the means
nk = sum(cc==k)
xsumk = sum(x[cc==k])
tau2.hat = 1/(nk/sigma[k]^2 + 1/tau^2)
mu.hat = tau2.hat*(xsumk/sigma[k]^2 + eta/tau^2)
mu[k] = rnorm(1, mu.hat, sqrt(tau2.hat))
# Sample the variances
dd.star = dd + nk/2
qq.star = qq + sum((x[cc==k] - mu[k])^2)/2
sigma[k] = sqrt(1/rgamma(1, dd.star, qq.star))
}
# Store samples
cc.out[s,] = cc
w.out[s] = w
mu.out[s,] = mu
sigma.out[s,] = sigma
logpost[s] = 0
for(i in 1:n){
if(cc[i] == 1){
logpost[s] = logpost[s] + log(w) + dnorm(x[i], mu[1], sigma[1], log=TRUE)
}else{
logpost[s] = logpost[s] + log(1-w) + dnorm(x[i], mu[2], sigma[2], log=TRUE)
}
}
logpost[s] = logpost[s] + dbeta(w, aa[1], aa[2], log=TRUE)
for(k in 1:2){
logpost[s] = logpost[s] + dnorm(mu[k], eta, tau, log = T) + dgamma(1/sigma[k]^2, dd, qq)/sigma[k]^4
}
if(s/500==floor(s/500)){
print(paste("s =",s))
}
}
## [1] "s = 500"
## [1] "s = 1000"
## [1] "s = 1500"
## [1] "s = 2000"
## [1] "s = 2500"
## [1] "s = 3000"
## [1] "s = 3500"
## [1] "s = 4000"
## [1] "s = 4500"
## [1] "s = 5000"
## [1] "s = 5500"
## [1] "s = 6000"
## [1] "s = 6500"
## [1] "s = 7000"
## [1] "s = 7500"
## [1] "s = 8000"
## [1] "s = 8500"
## [1] "s = 9000"
## [1] "s = 9500"
## [1] "s = 10000"
## [1] "s = 10500"
## [1] "s = 11000"
## [1] "s = 11500"
## [1] "s = 12000"
## Compute the samples of the density over a dense grid
xx = seq(0,7,length=150)
density.mcmc = array(0, dim=c(rrr-burn,length(xx)))
for(s in 1:(rrr-burn)){
density.mcmc[s,] = density.mcmc[s,] + w.out[s+burn]*dnorm(xx,mu.out[s+burn,1],sigma.out[s+burn,1]) +
(1-w.out[s+burn])*dnorm(xx,mu.out[s+burn,2],sigma.out[s+burn,2])
}
density.mcmc.m = apply(density.mcmc , 2, mean)
## Plot Bayesian estimate with pointwise credible bands
density.mcmc.lq = apply(density.mcmc, 2, quantile, 0.025)
density.mcmc.uq = apply(density.mcmc, 2, quantile, 0.975)
plot(xx, density.mcmc.m, type="n",ylim=c(0,max(density.mcmc.uq)),xlab="Eruptions", ylab="Density")
polygon(c(xx,rev(xx)), c(density.mcmc.lq, rev(density.mcmc.uq)), col="grey", border="grey")
lines(xx, density.mcmc.m, col="black", lwd=2)
points(x, rep(0,n))
Provide an MCMC algorithm to fit the two-component, location-and-scale mixture of Gaussians.
x = faithful$eruptions
KK = 2 # Based on the description of the dataset
n = length(x)
## Initialize the parameters
w = rep(1,KK)/KK
mu = rnorm(KK, mean(x), sd(x))
sigma = rep(1,KK)*sd(x)/KK
cc = sample(1:KK, n, replace=T, prob=w)
## Number of iterations of the sampler
rrr = 12000
burn = 2000
## Storing the samples
cc.out = array(0, dim=c(rrr, n))
w.out = array(0, dim=c(rrr, KK))
mu.out = array(0, dim=c(rrr, KK))
sigma.out = array(0, dim=c(rrr,KK))
logpost = rep(0, rrr)
#prior
eta = mean(x)
tau = sqrt(var(x))
dd = 2
qq = var(x)/KK
for(s in 1:rrr){ # Sample the indicators
for(i in 1:n){
v = rep(0,KK)
for(k in 1:KK){
v[k] = log(w[k]) + dnorm(x[i], mu[k], sigma[k], log=TRUE) #Compute the log of the weights }
v = exp(v - max(v))/sum(exp(v - max(v)))
cc[i] = sample(1:KK, 1, replace=TRUE, prob=v)
# Sample the weights
w[1] = rbeta(1,1+sum(cc==1),1+sum(cc==2))
w[2] = 1-w[1]
# Sample the means
for(k in 1:KK){
nk = sum(cc==k)
xsumk = sum(x[cc==k])
tau2.hat = 1/(nk/sigma[k]^2 + 1/tau^2)
mu.hat = tau2.hat*(xsumk/sigma[k]^2 + eta/tau^2)
mu[k] = rnorm(1, mu.hat, sqrt(tau2.hat))
}
# Sample the variances
dd.star = dd + n/2
for (k in 1:KK) {
qq.star = qq + sum((x[cc==k] - mu[k])^2)/2
sigma[k] = sqrt(1/rgamma(1, dd.star, qq.star))
}
# Store samples
cc.out[s,] = cc
w.out[s,] = w
mu.out[s,] = mu
sigma.out[s,] = sigma
for(i in 1:n){
logpost[s] = logpost[s] + log(w[cc[i]]) + dnorm(x[i], mu[cc[i]], sigma[cc[i]], log=TRUE)
}
logpost[s] = logpost[s] + dbeta(w[1],1,1,log = T)
for(k in 1:KK){
logpost[s] = logpost[s] + dnorm(mu[k], eta, tau, log=TRUE)+dgamma(1/sigma[k]^2, dd, qq, log=TRUE) - 4*log(sigma[k])
}
if(s/500==floor(s/500)){
print(paste("s =",s))
}
}
}
}
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 500"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1000"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 1500"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2000"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 2500"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3000"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 3500"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4000"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 4500"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5000"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 5500"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6000"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 6500"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7000"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 7500"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8000"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 8500"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9000"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 9500"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10000"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 10500"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11000"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 11500"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
## [1] "s = 12000"
Provide code to generate the density estimate on a grid consisting of 150 points in the interval [0,7].
xx = seq(0,7,length=150)
nxx = length(xx)
density.mcmc = array(0, dim=c(rrr-burn,length(xx)))
for(s in 1:(rrr-burn)){
for(k in 1:KK){
density.mcmc[s,] = density.mcmc[s,] + w.out[s+burn,k]*dnorm(xx,mu.out[s+burn,k],sigma.out[s+burn,k])
}
}
density.mcmc.m = apply(density.mcmc , 2, mean)
density.mcmc.lq = apply(density.mcmc, 2, quantile, 0.025)
density.mcmc.uq = apply(density.mcmc, 2, quantile, 0.975)
plot(xx, density.mcmc.m, type="n",ylim=c(0,max(density.mcmc.uq)),xlab="eruptions", ylab="Density") polygon(c(xx,rev(xx)), c(density.mcmc.lq, rev(density.mcmc.uq)), col="grey", border="grey")
lines(xx, density.mcmc.m, col="red", lwd=2)
points(x, rep(0,n))
## Error: <text>:15:100: unexpected symbol
## 14:
## 15: plot(xx, density.mcmc.m, type="n",ylim=c(0,max(density.mcmc.uq)),xlab="eruptions", ylab="Density") polygon
## ^
Provide the a graph of the density estimate on the interval [0,7].
Is the sampler for \(c\) correct?
The full conditional for \(c_i\) is given by,
\[ \Pr(c_i = 1|...) = \frac{\omega\frac{1}{\sqrt[]{2\pi\sigma_1}}exp\left\{-\frac{1}{2\sigma^2_1}(x_i-\mu_1)^2\right\}}{\omega\frac{1}{\sqrt[]{2\pi\sigma_1}}exp\left\{-\frac{1}{2\sigma^2_1}(x_i-\mu_1)^2\right\}+(1-\omega)\frac{1}{\sqrt[]{2\pi\sigma_2}}exp\left\{-\frac{1}{2\sigma^2_2}(x_i-\mu_2)^2\right\}} \]
(This assumes that the mixture weights are parameterized as
for(i in 1:n){
v = rep(0,2)
v[1] = log(w) + dnorm(x[i], mu[1], sigma[1], log=TRUE) #Compute the log of the weights
v[2] = log(1-w) + dnorm(x[i], mu[2], sigma[2], log=TRUE) #Compute the log of the weights
v = exp(v - max(v))/sum(exp(v - max(v)))
cc[i] = sample(1:2, 1, replace=TRUE, prob=v)
}
Is the sampler for \(\omega\) correct?
The full conditional is simply
\[ \omega \mid \cdots \mbox{Beta} \left(\alpha_1 + \sum_{i=1}^{n} \mathbf{1}(c_i=1), \alpha_2 + \sum_{i=1}^{n} \mathbf{1}(c_i=2)\right) \]
The following one line of code implements it:
w = rbeta(1, aa[1] + sum(cc==1), aa[2] + sum(cc==2))
Are the sampler for \(\mu\) correct?
\[ \mu_k \mid \cdots \mbox{Normal}\left( \left[ \frac{n_k}{\sigma_k^2} + \frac{1}{\tau^2} \right]^{-1}\left[ \frac{\sum_{i:c_i=k} x_i}{\sigma_k^2} + \frac{\eta}{\tau^2} \right], \left[ \frac{n_k}{\sigma_k^2} + \frac{1}{\tau^2} \right]^{-1}\right) \]
The following code implements the sampler:
# Sample the means
for(k in 1:2){
nk = sum(cc==k)
xsumk = sum(x[cc==k])
tau2.hat = 1/(nk/sigma[k]^2 + 1/tau^2)
mu.hat = tau2.hat*(xsumk/sigma[k]^2 + eta/tau^2)
mu[k] = rnorm(1, mu.hat, sqrt(tau2.hat))
}
Remember, however, that there are various ways in which the implementation could proceed.
Are the samplers for \(\sigma_k\) correct?
The full conditional takes the form:
\[ 1/\sigma^2_k \mid \cdots \mbox{Gamma}\left(d + \frac{n_k}{2} , q + \frac{1}{2} \sum_{i:c_k=k} (x_i - \mu_k)^2 \right) \]
The following code implements the sampler:
# Sample the variances
for(k in 1:2){
nk = sum(cc==k)
dd.star = dd + nk/2
qq.star = qq + sum((x[cc==k] - mu[k])^2)/2
sigma[k] = sqrt(1/rgamma(1, dd.star, qq.star))
}
Remember, however, that there are various ways in which the implementation could proceed.
Does the code to generate the density estimate appear correct?
This is an example that, as before, assumes that the weights are parameterized as \(\omega\) and \(1- \omega\).
## Plot Bayesian estimate with pointwise credible bands
density.mcmc.lq = apply(density.mcmc, 2, quantile, 0.025)
density.mcmc.uq = apply(density.mcmc, 2, quantile, 0.975)
plot(xx, density.mcmc.m, type="n",ylim=c(0,max(density.mcmc.uq)),xlab="Eruptions", ylab="Density")
polygon(c(xx,rev(xx)), c(density.mcmc.lq, rev(density.mcmc.uq)), col="grey", border="grey")
lines(xx, density.mcmc.m, col="black", lwd=2)
points(x, rep(0,n))
Note that this will generate not only a point estimate, but also pointwise credible intervals.
Does the graph appear to be correct?
The density estimate should look similar to the Figure below. Note that the variance of both components is clearly different.
The full code for the MCMC algorithm follows:
x = faithful$eruptions
n = length(x)
plot(density(x))
points(x,rep(0,n))
## Priors set up using an "empirical Bayes" approach
aa = rep(1,2)
eta = mean(x)
tau = sqrt(var(x))
dd = 2
qq = var(x)/2
## Initialize the parameters
w = 1/2
mu = rnorm(2, mean(x), sd(x))
sigma = rep(sd(x)/2,2)
cc = sample(1:2, n, replace=T, prob=c(1/2,1/2))
## Number of iterations of the sampler
rrr = 12000
burn = 2000
## Storing the samples
cc.out = array(0, dim=c(rrr, n))
w.out = rep(0, rrr)
mu.out = array(0, dim=c(rrr, 2))
sigma.out = array(0, dim=c(rrr, 2))
logpost = rep(0, rrr)
for(s in 1:rrr){
# Sample the indicators
for(i in 1:n){
v = rep(0,2)
v[1] = log(w) + dnorm(x[i], mu[1], sigma[1], log=TRUE) #Compute the log of the weights
v[2] = log(1-w) + dnorm(x[i], mu[2], sigma[2], log=TRUE) #Compute the log of the weights
v = exp(v - max(v))/sum(exp(v - max(v)))
cc[i] = sample(1:2, 1, replace=TRUE, prob=v)
}
# Sample the weights
w = rbeta(1, aa[1] + sum(cc==1), aa[2] + sum(cc==2))
# Sample the parameters of the components
for(k in 1:2){
# Sample the means
nk = sum(cc==k)
xsumk = sum(x[cc==k])
tau2.hat = 1/(nk/sigma[k]^2 + 1/tau^2)
mu.hat = tau2.hat*(xsumk/sigma[k]^2 + eta/tau^2)
mu[k] = rnorm(1, mu.hat, sqrt(tau2.hat))
# Sample the variances
dd.star = dd + nk/2
qq.star = qq + sum((x[cc==k] - mu[k])^2)/2
sigma[k] = sqrt(1/rgamma(1, dd.star, qq.star))
}
# Store samples
cc.out[s,] = cc
w.out[s] = w
mu.out[s,] = mu
sigma.out[s,] = sigma
logpost[s] = 0
for(i in 1:n){
if(cc[i] == 1){
logpost[s] = logpost[s] + log(w) + dnorm(x[i], mu[1], sigma[1], log=TRUE)
}else{
logpost[s] = logpost[s] + log(1-w) + dnorm(x[i], mu[2], sigma[2], log=TRUE)
}
}
logpost[s] = logpost[s] + dbeta(w, aa[1], aa[2], log=TRUE)
for(k in 1:2){
logpost[s] = logpost[s] + dnorm(mu[k], eta, tau, log = T) + dgamma(1/sigma[k]^2, dd, qq)/sigma[k]^4
}
if(s/500==floor(s/500)){
print(paste("s =",s))
}
}
## [1] "s = 500"
## [1] "s = 1000"
## [1] "s = 1500"
## [1] "s = 2000"
## [1] "s = 2500"
## [1] "s = 3000"
## [1] "s = 3500"
## [1] "s = 4000"
## [1] "s = 4500"
## [1] "s = 5000"
## [1] "s = 5500"
## [1] "s = 6000"
## [1] "s = 6500"
## [1] "s = 7000"
## [1] "s = 7500"
## [1] "s = 8000"
## [1] "s = 8500"
## [1] "s = 9000"
## [1] "s = 9500"
## [1] "s = 10000"
## [1] "s = 10500"
## [1] "s = 11000"
## [1] "s = 11500"
## [1] "s = 12000"
## Compute the samples of the density over a dense grid
xx = seq(0,7,length=150)
density.mcmc = array(0, dim=c(rrr-burn,length(xx)))
for(s in 1:(rrr-burn)){
density.mcmc[s,] = density.mcmc[s,] + w.out[s+burn]*dnorm(xx,mu.out[s+burn,1],sigma.out[s+burn,1]) +
(1-w.out[s+burn])*dnorm(xx,mu.out[s+burn,2],sigma.out[s+burn,2])
}
density.mcmc.m = apply(density.mcmc , 2, mean)
## Plot Bayesian estimate with pointwise credible bands
density.mcmc.lq = apply(density.mcmc, 2, quantile, 0.025)
density.mcmc.uq = apply(density.mcmc, 2, quantile, 0.975)
plot(xx, density.mcmc.m, type="n",ylim=c(0,max(density.mcmc.uq)),xlab="Eruptions", ylab="Density")
polygon(c(xx,rev(xx)), c(density.mcmc.lq, rev(density.mcmc.uq)), col="grey", border="grey")
lines(xx, density.mcmc.m, col="black", lwd=2)
points(x, rep(0,n))
Provide an MCMC algorithm to fit the two-component, location-and-scale mixture of Gaussians.
Provide code to generate the density estimate on a grid consisting of 150 points in the interval [0,7].
Provide the a graph of the density estimate on the interval [0,7].
Is the sampler for \(c\) correct?
The full conditional for \(c_i\) is given by,
\[ \Pr(c_i = 1|...) = \frac{\omega\frac{1}{\sqrt[]{2\pi\sigma_1}}exp\left\{-\frac{1}{2\sigma^2_1}(x_i-\mu_1)^2\right\}}{\omega\frac{1}{\sqrt[]{2\pi\sigma_1}}exp\left\{-\frac{1}{2\sigma^2_1}(x_i-\mu_1)^2\right\}+(1-\omega)\frac{1}{\sqrt[]{2\pi\sigma_2}}exp\left\{-\frac{1}{2\sigma^2_2}(x_i-\mu_2)^2\right\}} \]
(This assumes that the mixture weights are parameterized as
for(i in 1:n){
v = rep(0,2)
v[1] = log(w) + dnorm(x[i], mu[1], sigma[1], log=TRUE) #Compute the log of the weights
v[2] = log(1-w) + dnorm(x[i], mu[2], sigma[2], log=TRUE) #Compute the log of the weights
v = exp(v - max(v))/sum(exp(v - max(v)))
cc[i] = sample(1:2, 1, replace=TRUE, prob=v)
}
Is the sampler for \(\omega\) correct?
The full conditional is simply
\[ \omega \mid \cdots \mbox{Beta} \left(\alpha_1 + \sum_{i=1}^{n} \mathbf{1}(c_i=1), \alpha_2 + \sum_{i=1}^{n} \mathbf{1}(c_i=2)\right) \]
The following one line of code implements it:
w = rbeta(1, aa[1] + sum(cc==1), aa[2] + sum(cc==2))
Are the sampler for \(\mu\) correct?
\[ \mu_k \mid \cdots \mbox{Normal}\left( \left[ \frac{n_k}{\sigma_k^2} + \frac{1}{\tau^2} \right]^{-1}\left[ \frac{\sum_{i:c_i=k} x_i}{\sigma_k^2} + \frac{\eta}{\tau^2} \right], \left[ \frac{n_k}{\sigma_k^2} + \frac{1}{\tau^2} \right]^{-1}\right) \]
The following code implements the sampler:
# Sample the means
for(k in 1:2){
nk = sum(cc==k)
xsumk = sum(x[cc==k])
tau2.hat = 1/(nk/sigma[k]^2 + 1/tau^2)
mu.hat = tau2.hat*(xsumk/sigma[k]^2 + eta/tau^2)
mu[k] = rnorm(1, mu.hat, sqrt(tau2.hat))
}
Remember, however, that there are various ways in which the implementation could proceed.
Are the samplers for \(\sigma_k\) correct?
The full conditional takes the form:
\[ 1/\sigma^2_k \mid \cdots \mbox{Gamma}\left(d + \frac{n_k}{2} , q + \frac{1}{2} \sum_{i:c_k=k} (x_i - \mu_k)^2 \right) \]
The following code implements the sampler:
# Sample the variances
for(k in 1:2){
nk = sum(cc==k)
dd.star = dd + nk/2
qq.star = qq + sum((x[cc==k] - mu[k])^2)/2
sigma[k] = sqrt(1/rgamma(1, dd.star, qq.star))
}
Remember, however, that there are various ways in which the implementation could proceed.
Does the code to generate the density estimate appear correct?
This is an example that, as before, assumes that the weights are parameterized as \(\omega\) and \(1- \omega\).
## Plot Bayesian estimate with pointwise credible bands
density.mcmc.lq = apply(density.mcmc, 2, quantile, 0.025)
density.mcmc.uq = apply(density.mcmc, 2, quantile, 0.975)
plot(xx, density.mcmc.m, type="n",ylim=c(0,max(density.mcmc.uq)),xlab="Eruptions", ylab="Density")
polygon(c(xx,rev(xx)), c(density.mcmc.lq, rev(density.mcmc.uq)), col="grey", border="grey")
lines(xx, density.mcmc.m, col="black", lwd=2)
points(x, rep(0,n))
Note that this will generate not only a point estimate, but also pointwise credible intervals.
Does the graph appear to be correct?
The density estimate should look similar to the Figure below. Note that the variance of both components is clearly different.
The full code for the MCMC algorithm follows:
x = faithful$eruptions
n = length(x)
plot(density(x))
points(x,rep(0,n))
## Priors set up using an "empirical Bayes" approach
aa = rep(1,2)
eta = mean(x)
tau = sqrt(var(x))
dd = 2
qq = var(x)/2
## Initialize the parameters
w = 1/2
mu = rnorm(2, mean(x), sd(x))
sigma = rep(sd(x)/2,2)
cc = sample(1:2, n, replace=T, prob=c(1/2,1/2))
## Number of iterations of the sampler
rrr = 12000
burn = 2000
## Storing the samples
cc.out = array(0, dim=c(rrr, n))
w.out = rep(0, rrr)
mu.out = array(0, dim=c(rrr, 2))
sigma.out = array(0, dim=c(rrr, 2))
logpost = rep(0, rrr)
for(s in 1:rrr){
# Sample the indicators
for(i in 1:n){
v = rep(0,2)
v[1] = log(w) + dnorm(x[i], mu[1], sigma[1], log=TRUE) #Compute the log of the weights
v[2] = log(1-w) + dnorm(x[i], mu[2], sigma[2], log=TRUE) #Compute the log of the weights
v = exp(v - max(v))/sum(exp(v - max(v)))
cc[i] = sample(1:2, 1, replace=TRUE, prob=v)
}
# Sample the weights
w = rbeta(1, aa[1] + sum(cc==1), aa[2] + sum(cc==2))
# Sample the parameters of the components
for(k in 1:2){
# Sample the means
nk = sum(cc==k)
xsumk = sum(x[cc==k])
tau2.hat = 1/(nk/sigma[k]^2 + 1/tau^2)
mu.hat = tau2.hat*(xsumk/sigma[k]^2 + eta/tau^2)
mu[k] = rnorm(1, mu.hat, sqrt(tau2.hat))
# Sample the variances
dd.star = dd + nk/2
qq.star = qq + sum((x[cc==k] - mu[k])^2)/2
sigma[k] = sqrt(1/rgamma(1, dd.star, qq.star))
}
# Store samples
cc.out[s,] = cc
w.out[s] = w
mu.out[s,] = mu
sigma.out[s,] = sigma
logpost[s] = 0
for(i in 1:n){
if(cc[i] == 1){
logpost[s] = logpost[s] + log(w) + dnorm(x[i], mu[1], sigma[1], log=TRUE)
}else{
logpost[s] = logpost[s] + log(1-w) + dnorm(x[i], mu[2], sigma[2], log=TRUE)
}
}
logpost[s] = logpost[s] + dbeta(w, aa[1], aa[2], log=TRUE)
for(k in 1:2){
logpost[s] = logpost[s] + dnorm(mu[k], eta, tau, log = T) + dgamma(1/sigma[k]^2, dd, qq)/sigma[k]^4
}
if(s/500==floor(s/500)){
print(paste("s =",s))
}
}
## [1] "s = 500"
## [1] "s = 1000"
## [1] "s = 1500"
## [1] "s = 2000"
## [1] "s = 2500"
## [1] "s = 3000"
## [1] "s = 3500"
## [1] "s = 4000"
## [1] "s = 4500"
## [1] "s = 5000"
## [1] "s = 5500"
## [1] "s = 6000"
## [1] "s = 6500"
## [1] "s = 7000"
## [1] "s = 7500"
## [1] "s = 8000"
## [1] "s = 8500"
## [1] "s = 9000"
## [1] "s = 9500"
## [1] "s = 10000"
## [1] "s = 10500"
## [1] "s = 11000"
## [1] "s = 11500"
## [1] "s = 12000"
## Compute the samples of the density over a dense grid
xx = seq(0,7,length=150)
density.mcmc = array(0, dim=c(rrr-burn,length(xx)))
for(s in 1:(rrr-burn)){
density.mcmc[s,] = density.mcmc[s,] + w.out[s+burn]*dnorm(xx,mu.out[s+burn,1],sigma.out[s+burn,1]) +
(1-w.out[s+burn])*dnorm(xx,mu.out[s+burn,2],sigma.out[s+burn,2])
}
density.mcmc.m = apply(density.mcmc , 2, mean)
## Plot Bayesian estimate with pointwise credible bands
density.mcmc.lq = apply(density.mcmc, 2, quantile, 0.025)
density.mcmc.uq = apply(density.mcmc, 2, quantile, 0.975)
plot(xx, density.mcmc.m, type="n",ylim=c(0,max(density.mcmc.uq)),xlab="Eruptions", ylab="Density")
polygon(c(xx,rev(xx)), c(density.mcmc.lq, rev(density.mcmc.uq)), col="grey", border="grey")
lines(xx, density.mcmc.m, col="black", lwd=2)
points(x, rep(0,n))
Provide an MCMC algorithm to fit the two-component, location-and-scale mixture of Gaussians.
Provide code to generate the density estimate on a grid consisting of 150 points in the interval [0,7].
Provide the a graph of the density estimate on the interval [0,7].
Is the sampler for \(c\) correct?
The full conditional for \(c_i\) is given by,
\[ \Pr(c_i = 1|...) = \frac{\omega\frac{1}{\sqrt[]{2\pi\sigma_1}}exp\left\{-\frac{1}{2\sigma^2_1}(x_i-\mu_1)^2\right\}}{\omega\frac{1}{\sqrt[]{2\pi\sigma_1}}exp\left\{-\frac{1}{2\sigma^2_1}(x_i-\mu_1)^2\right\}+(1-\omega)\frac{1}{\sqrt[]{2\pi\sigma_2}}exp\left\{-\frac{1}{2\sigma^2_2}(x_i-\mu_2)^2\right\}} \]
(This assumes that the mixture weights are parameterized as
for(i in 1:n){
v = rep(0,2)
v[1] = log(w) + dnorm(x[i], mu[1], sigma[1], log=TRUE) #Compute the log of the weights
v[2] = log(1-w) + dnorm(x[i], mu[2], sigma[2], log=TRUE) #Compute the log of the weights
v = exp(v - max(v))/sum(exp(v - max(v)))
cc[i] = sample(1:2, 1, replace=TRUE, prob=v)
}
Is the sampler for \(\omega\) correct?
The full conditional is simply
\[ \omega \mid \cdots \mbox{Beta} \left(\alpha_1 + \sum_{i=1}^{n} \mathbf{1}(c_i=1), \alpha_2 + \sum_{i=1}^{n} \mathbf{1}(c_i=2)\right) \]
The following one line of code implements it:
w = rbeta(1, aa[1] + sum(cc==1), aa[2] + sum(cc==2))
Are the sampler for \(\mu\) correct?
\[ \mu_k \mid \cdots \mbox{Normal}\left( \left[ \frac{n_k}{\sigma_k^2} + \frac{1}{\tau^2} \right]^{-1}\left[ \frac{\sum_{i:c_i=k} x_i}{\sigma_k^2} + \frac{\eta}{\tau^2} \right], \left[ \frac{n_k}{\sigma_k^2} + \frac{1}{\tau^2} \right]^{-1}\right) \]
The following code implements the sampler:
# Sample the means
for(k in 1:2){
nk = sum(cc==k)
xsumk = sum(x[cc==k])
tau2.hat = 1/(nk/sigma[k]^2 + 1/tau^2)
mu.hat = tau2.hat*(xsumk/sigma[k]^2 + eta/tau^2)
mu[k] = rnorm(1, mu.hat, sqrt(tau2.hat))
}
Remember, however, that there are various ways in which the implementation could proceed.
Are the samplers for \(\sigma_k\) correct?
The full conditional takes the form:
\[ 1/\sigma^2_k \mid \cdots \mbox{Gamma}\left(d + \frac{n_k}{2} , q + \frac{1}{2} \sum_{i:c_k=k} (x_i - \mu_k)^2 \right) \]
The following code implements the sampler:
# Sample the variances
for(k in 1:2){
nk = sum(cc==k)
dd.star = dd + nk/2
qq.star = qq + sum((x[cc==k] - mu[k])^2)/2
sigma[k] = sqrt(1/rgamma(1, dd.star, qq.star))
}
Remember, however, that there are various ways in which the implementation could proceed.
Does the code to generate the density estimate appear correct?
This is an example that, as before, assumes that the weights are parameterized as \(\omega\) and \(1- \omega\).
## Plot Bayesian estimate with pointwise credible bands
density.mcmc.lq = apply(density.mcmc, 2, quantile, 0.025)
density.mcmc.uq = apply(density.mcmc, 2, quantile, 0.975)
plot(xx, density.mcmc.m, type="n",ylim=c(0,max(density.mcmc.uq)),xlab="Eruptions", ylab="Density")
polygon(c(xx,rev(xx)), c(density.mcmc.lq, rev(density.mcmc.uq)), col="grey", border="grey")
lines(xx, density.mcmc.m, col="black", lwd=2)
points(x, rep(0,n))
Note that this will generate not only a point estimate, but also pointwise credible intervals.
Does the graph appear to be correct?
The density estimate should look similar to the Figure below. Note that the variance of both components is clearly different.
The full code for the MCMC algorithm follows:
x = faithful$eruptions
n = length(x)
plot(density(x))
points(x,rep(0,n))
## Priors set up using an "empirical Bayes" approach
aa = rep(1,2)
eta = mean(x)
tau = sqrt(var(x))
dd = 2
qq = var(x)/2
## Initialize the parameters
w = 1/2
mu = rnorm(2, mean(x), sd(x))
sigma = rep(sd(x)/2,2)
cc = sample(1:2, n, replace=T, prob=c(1/2,1/2))
## Number of iterations of the sampler
rrr = 12000
burn = 2000
## Storing the samples
cc.out = array(0, dim=c(rrr, n))
w.out = rep(0, rrr)
mu.out = array(0, dim=c(rrr, 2))
sigma.out = array(0, dim=c(rrr, 2))
logpost = rep(0, rrr)
for(s in 1:rrr){
# Sample the indicators
for(i in 1:n){
v = rep(0,2)
v[1] = log(w) + dnorm(x[i], mu[1], sigma[1], log=TRUE) #Compute the log of the weights
v[2] = log(1-w) + dnorm(x[i], mu[2], sigma[2], log=TRUE) #Compute the log of the weights
v = exp(v - max(v))/sum(exp(v - max(v)))
cc[i] = sample(1:2, 1, replace=TRUE, prob=v)
}
# Sample the weights
w = rbeta(1, aa[1] + sum(cc==1), aa[2] + sum(cc==2))
# Sample the parameters of the components
for(k in 1:2){
# Sample the means
nk = sum(cc==k)
xsumk = sum(x[cc==k])
tau2.hat = 1/(nk/sigma[k]^2 + 1/tau^2)
mu.hat = tau2.hat*(xsumk/sigma[k]^2 + eta/tau^2)
mu[k] = rnorm(1, mu.hat, sqrt(tau2.hat))
# Sample the variances
dd.star = dd + nk/2
qq.star = qq + sum((x[cc==k] - mu[k])^2)/2
sigma[k] = sqrt(1/rgamma(1, dd.star, qq.star))
}
# Store samples
cc.out[s,] = cc
w.out[s] = w
mu.out[s,] = mu
sigma.out[s,] = sigma
logpost[s] = 0
for(i in 1:n){
if(cc[i] == 1){
logpost[s] = logpost[s] + log(w) + dnorm(x[i], mu[1], sigma[1], log=TRUE)
}else{
logpost[s] = logpost[s] + log(1-w) + dnorm(x[i], mu[2], sigma[2], log=TRUE)
}
}
logpost[s] = logpost[s] + dbeta(w, aa[1], aa[2], log=TRUE)
for(k in 1:2){
logpost[s] = logpost[s] + dnorm(mu[k], eta, tau, log = T) + dgamma(1/sigma[k]^2, dd, qq)/sigma[k]^4
}
if(s/500==floor(s/500)){
print(paste("s =",s))
}
}
## [1] "s = 500"
## [1] "s = 1000"
## [1] "s = 1500"
## [1] "s = 2000"
## [1] "s = 2500"
## [1] "s = 3000"
## [1] "s = 3500"
## [1] "s = 4000"
## [1] "s = 4500"
## [1] "s = 5000"
## [1] "s = 5500"
## [1] "s = 6000"
## [1] "s = 6500"
## [1] "s = 7000"
## [1] "s = 7500"
## [1] "s = 8000"
## [1] "s = 8500"
## [1] "s = 9000"
## [1] "s = 9500"
## [1] "s = 10000"
## [1] "s = 10500"
## [1] "s = 11000"
## [1] "s = 11500"
## [1] "s = 12000"
## Compute the samples of the density over a dense grid
xx = seq(0,7,length=150)
density.mcmc = array(0, dim=c(rrr-burn,length(xx)))
for(s in 1:(rrr-burn)){
density.mcmc[s,] = density.mcmc[s,] + w.out[s+burn]*dnorm(xx,mu.out[s+burn,1],sigma.out[s+burn,1]) +
(1-w.out[s+burn])*dnorm(xx,mu.out[s+burn,2],sigma.out[s+burn,2])
}
density.mcmc.m = apply(density.mcmc , 2, mean)
## Plot Bayesian estimate with pointwise credible bands
density.mcmc.lq = apply(density.mcmc, 2, quantile, 0.025)
density.mcmc.uq = apply(density.mcmc, 2, quantile, 0.975)
plot(xx, density.mcmc.m, type="n",ylim=c(0,max(density.mcmc.uq)),xlab="Eruptions", ylab="Density")
polygon(c(xx,rev(xx)), c(density.mcmc.lq, rev(density.mcmc.uq)), col="grey", border="grey")
lines(xx, density.mcmc.m, col="black", lwd=2)
points(x, rep(0,n))
It’s useful to record some information about how your file was created.
suppressMessages(require('dplyr', quietly = TRUE))
suppressMessages(require('magrittr', quietly = TRUE))
suppressMessages(require('formattable', quietly = TRUE))
suppressMessages(require('knitr', quietly = TRUE))
suppressMessages(require('kableExtra', quietly = TRUE))
sys1 <- devtools::session_info()$platform %>%
unlist %>% data.frame(Category = names(.), session_info = .)
rownames(sys1) <- NULL
sys2 <- data.frame(Sys.info()) %>%
dplyr::mutate(Category = rownames(.)) %>% .[2:1]
names(sys2)[2] <- c('Sys.info')
rownames(sys2) <- NULL
if (nrow(sys1) == 9 & nrow(sys2) == 8) {
sys2 %<>% rbind(., data.frame(
Category = 'Current time',
Sys.info = paste(as.character(lubridate::now('Asia/Tokyo')), 'JST🗾')))
} else {
sys1 %<>% rbind(., data.frame(
Category = 'Current time',
session_info = paste(as.character(lubridate::now('Asia/Tokyo')), 'JST🗾')))
}
sys <- cbind(sys1, sys2) %>%
kbl(caption = 'Additional session information:') %>%
kable_styling(bootstrap_options = c('striped', 'hover', 'condensed', 'responsive')) %>%
row_spec(0, background = 'DimGrey', color = 'yellow') %>%
column_spec(1, background = 'CornflowerBlue', color = 'red') %>%
column_spec(2, background = 'grey', color = 'black') %>%
column_spec(3, background = 'CornflowerBlue', color = 'blue') %>%
column_spec(4, background = 'grey', color = 'white') %>%
row_spec(9, bold = T, color = 'yellow', background = '#D7261E')
rm(sys1, sys2)
sys
| Category | session_info | Category | Sys.info |
|---|---|---|---|
| version | R version 4.1.0 (2021-05-18) | sysname | Linux |
| os | Ubuntu 20.04.2 LTS | release | 5.8.0-54-generic |
| system | x86_64, linux-gnu | version | #61~20.04.1-Ubuntu SMP Thu May 13 00:05:49 UTC 2021 |
| ui | X11 | nodename | Scibrokes-Trading |
| language | en | machine | x86_64 |
| collate | en_US.UTF-8 | login | englianhu |
| ctype | en_US.UTF-8 | user | englianhu |
| tz | Asia/Tokyo | effective_user | englianhu |
| date | 2021-05-24 | Current time | 2021-05-24 19:19:20 JST🗾 |